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Chapter 14 Arithmetic and Geometric Sequences
Chapter 14
14.1
Arithmetic and Geometric
Sequences
WARM-UP EXERCISE
1. (a) If T(n)  n  3, find T(1), T(2), T(3) and T(4).
(b) If T(n)  4  2n, find T(1), T(2), T(3) and T(4).
2. (a) If T (n)  2(3) n 1 , find T(1), T(2), T(3) and T(4).
1
(b) If T (n)  3( ) n 1 , find T(1), T(2), T(3) and T(4).
2
3. If T ( n)  n 2  2n ,
(a) find T(6)  T(5).
(b) find T(10)  T(9).
4. If S(n)  2n(n  3),
(a) find S(n  2).
(b) find S(n  3).
5. Find the value of n in each of the following and correct your answers to 1 decimal place.
2
1
(a) 100 (2) n  10
(b) 5( ) n 
3
25
BUILD-UP EXERCISE
Exercise 14A
 Elementary Set
Level 1

Write down the first 3 terms of the sequences of which the general terms are as follows. ( 1  4)
1. (a) T(n)  n  6
(c) T(n)  70  4n
(b) T(n)  5  2n
Ex.14A Elementary Set
[ This part provides three extra sets of questions for each exercise in the textbook, namely Elementary Set ,
Intermediate Set and Advanced Set. You may choose to complete any ONE set according to your need. ]
14.2
New Trend Mathematics S5A — Supplement
2. (a) T(n)  n 2  1
3n 2  4
(c) T(n) 
2
(b) T(n)  2n 2  5
3. (a) T(n)  (n  1)2
(b) T(n)  (2n  1)2
(c) T(n)  n 2  3n
4. (a) T(n)  4 n
(b) T(n)  (3) n
(c) T(n)  (1) n1
Find the general term T(n) of each of the following sequences. (5  7)
5. (a) 5, 6, 7, 8, 
(b) 6, 12, 18, 24, 
(c) 2, 5, 8, 11, 
6. (a)
1, 3, 5, 7 , 
(b) 1 4, 2  5, 3  6, 4  7, 
Ex.14A Elementary Set
(c) 2  5, 3  6, 4  7, 5  8, 
7. (a)
1 1 1 1
, , , ,
4 8 12 16
(b)
3 4 5 6
, , , ,
2 3 4 5
(c) 5, 25, 125, 625, 
8. It is given that T(n)  3n  7.
(a) Find T(2)  T(1).
9. It is given that T(n)  4  5n .
T ( 2)
(a) Find
.
T (1)
(b) Find T(5)  T(4).
(b) Find
T ( 4)
.
T (3)
10. It is given that T(n)  2 n1  3 n .
(a) Find T(1).
T ( 2)
(c) Find
.
T (1)
11. It is given that T(n)  n 2  n.
(a) Find T(2).
(b) Find T(5).
(c) Find T(k  1).
(b) Find T(2).
T ( 5)
(d) Find
.
T ( 4)
Chapter 14 Arithmetic and Geometric Sequences
14.3
12. It is given that T(n)  pn  7 where p is a constant and T(9)  34. Find the value of p.
13. The following is a sequence of figures formed by squares of equal area.
,
1st figure
,
2nd figure
,


3rd figure
Ex.14A Elementary Set
Level 2
(a) Find the respective numbers of squares in the 4th, 5th and 6th figure s.
(b) Find the number of squares in the 10th figure.
 Intermediate Set
Level 1

Write down the first 5 terms of the sequences of which the general terms are as follows. ( 14  16)
14. (a) T(n)  5n  3
(b) T(n)  10  3n
(c) T(n)  60  7n
15. (a) T(n)  3n 2  4
(b) T(n)  (n  2) 2
16. (a) T(n)  (1) n1  2 n
(c) T(n)  (1) n1  (1) n2
(b) T (n) 
(1) n 1
2n  3
Find the general term T(n) of each of the following sequences. (17  18)
17. (a)
1, 4 , 7 , 10 , 
(b) 1 5, 2  6, 3  7, 4  8, 
(c) 1 6, 3  7, 5  8, 7  9, 
3 5 7
, , ,
2 4 6
2 3
(c) 1, ,
,
7 49
18. (a)
9
,
8
4
,
343
(b) 3, 12, 48, 192, 
19. It is given that T(n)  7n  4.
(a) Find T(2)  T(1).
(c) Find T(n  1)  T(n).
(b) Find T(12)  T(11).
Ex.14A Intermediate Set
(c) T(n)  2n 2  n
14.4
New Trend Mathematics S5A — Supplement
20. It is given that T(n)  3  4n1 .
T ( 2)
(a) Find
.
T (1)
T (n  1)
(c) Find
.
T ( n)
(b) Find
T ( 6)
.
T ( 5)
Level 2
21. It is given that T(n)  3 n1  4 n .
(a) Find T(1).
T (5)
(c) Find
.
T (3)
(b) Find T(2).
T (n  1)
(d) Find
.
T ( n)
22. It is given that T(n)  pn  6 where p is a constant and T(7)  34.
Ex.14A Intermediate Set
(a) Find the value of p.
(b) Find T(n)  T(n  1).
23. It is given that T(n)  an 2  bn  3 where a and b are constants. If T(5)  28 and T(10)  153,
(a) find the values of a and b.
(b) find the first and second terms.
24. The following is a sequence of figures formed by triangles of equal area.
,
1st figure
2nd figure

,
,
3rd figure
(a) Find the respective numbers of triangles in the 4th, 5th and 6th figures.
(b) Find the number of triangles in the 8th figure.
25. In a concert hall, the number of seats in the nth row is T(n) where T(n)  12  3n.
(a) Find the number of seats in the 11th row.
(b) A group of 90 students are asked to fill in three consecutive rows completely. Is it
possible? If yes, which three rows will they sit?
Ex.14A Advanced Set
 Advanced Set
Level 1

Find the general term T(n) of each of the following sequences. (26  27)
26. (a)
2 , 6 , 18 , 54 , 
(c) 3  4, 5  7, 7 10, 9 13, 
(b) 2  7, 3  9, 4  11, 5  13, 
Chapter 14 Arithmetic and Geometric Sequences
5 7 9 11
, , , ,
4 9 14 19
1
1 1
1
(c) ,  ,
,
,
2 10 50
250
27. (a)
14.5
(b) 11, 101, 1 001, 10 001, 
28. It is given that T(n)  (n  2) 2  1.
(a) Find T(2)  T(1).
(b) Find T(20)  T(19).
(c) Find T(n  1)  T(n).
29. It is given that T(n)  7  3n2 .
T ( 2)
(a) Find
.
T (1)
T (n  1)
(c) Find
.
T ( n)
(b) Find
T (11)
.
T (10 )
30. It is given that T(n)  3 n1  6 n2 .
(a) Find T(1).
(c) Find
T (7 )
.
T ( 5)
1
31. It is given that T (n)  25  ( ) n 1 .
5
T ( 2)
(a) Find
.
T (3)
T ( n  2)
(c) Find
.
T ( n)
(b) Find T(2).
(d) Find
T (n  1)
.
T ( n)
T (7 )
.
T ( 4)
T ( n  1)
(d) Find
.
T ( n  2)
(b) Find
32. It is given that T(n)  pn  9 where p is a constant and T(5)  39.
(a) Find the value of p.
(b) Find T(n)  T(n  1).
33. It is given that T(n)  an 2  bn  18 where a and b are constants. If T(3)  33 and T(8)  438,
(a) find the values of a and b.
(b) find the first and second terms.
34. The general term T(n) of the sequence 2, 3, 6, 11,  is given by T(n)  an 2  bn  3, where a
and b are constants.
(a) Find the values of a and b.
(b) Find T(10).
Ex.14A Advanced Set
Level 2
14.6
New Trend Mathematics S5A — Supplement
35. The following is a sequence of figures formed by squares with areas of 2 square units each.
,
1st figure
,
,
2nd figure

3rd figure
Ex.14A Advanced Set
(a) Find the respective areas of the 4th, 5th and 6th figures.
(b) Find the area of the 10th figure.
36. In a lecture hall, the number of seats in the nth row is T(n) where T(n)  15  4n.
(a) Find the number of seats in the 16th row.
(b) A lecturer would like to ask the 172 students who are taking his course to fill in four
consecutive rows completely. Is it possible? Explain your answer briefly.
37. It is known that for a sequence T(n),
(a) Express T(n) in terms of T(n  1).
T ( n)
 T (n)  T (n  1) and T(1)  2.
T (n  1)
(b) Hence find T(2) and T(3).
Exercise 14B
 Elementary Set
Level 1

1. Determine whether each of the following is an arithmetic sequence. If it is so, find the
common difference of each arithmetic sequence.
(a) 6, 11, 16, 21, 
Ex.14B Elementary Set
(c) 8, 6.6, 5.2, 3.8, 
(b) 4, 9, 16, 25, 
1 1 3
(d) , , , 1, 
4 2 4
2. Write down the general term of each of the following arithmetic sequences.
(a) 1st term  1, common difference  4
(b) 1st term  10, common difference  3
(c) 1st term  6, common difference  5
(d) 1st term  9, common difference  1
3. For each of the following general terms T(n), find the 1st term and the common difference.
(a) T(n)  n  10
(b) T(n)  3(n  1)
(c) T(n)  4  3n
(d) T(n)  2n  0.5
Chapter 14 Arithmetic and Geometric Sequences
14.7
4. Find the general term T(n) of each of the following arithmetic sequences.
(a) 17, 21, 25, 29, 
(b) 45, 35, 25, 15, 
(c) 0.5, 0,  0.5, 1, 
(d) 3, 4.5, 6, 7.5, 
5. Consider the arithmetic sequence 92, 79, 66, 53,  .
(a) Find the general term.
(b) Find the 20th term.
6. Consider the arithmetic sequence 6, 13, 20, 27,  .
(a) Find the general term.
(b) Find the 18th term.
(c) If the nth term is 153, find the value of n.
8. Insert a number between 14 and 40 such that these three numbers form an arithmetic
sequence.
9. Insert two numbers between 25 and 8 such that these four numbers form an arithmetic
sequence.
10. Find the number of terms in each of the following arithmetic sequences.
(a) 2, 5, 8, 11,  , 80
(b) 17, 15, 13, 11,  , 17
(c) 32, 38, 44, 50,  , 116
(d) 9, 11.5, 14, 16.5,  , 129
11. The common difference of an arithmetic sequence is 3. The 7th term is 64. Find the general
term of the arithmetic sequence.
12. In the figure, AOB is a straight line. x, y and z form an
arithmetic sequence. If x  37, find z.
y
z
x
A
13. Consider the arithmetic sequence x  6, 14  x, 2  x,  .
(a) Find the value of x.
(b) Find the general term of the arithmetic sequence.
O
B
Ex.14B Elementary Set
7. If 25, a, b and 1 form an arithmetic sequence, find the values of a and b.
14.8
New Trend Mathematics S5A — Supplement
14. 5  x, 7  3x, 11  4x,  is an arithmetic sequence.
(a) Find the value of x.
(b) Find the general term of the arithmetic sequence.
(c) Find the 12th term of the arithmetic sequence.
Level 2
15. The 4th term and the 6th term of an arithmetic sequence are 15 and 23 respectively.
(a) Find the 1st term and the common difference of the arithmetic sequence.
(b) Find the general term of the arithmetic sequence.
Ex.14B Elementary Set
16. The 7th term and the 10th term of an arithmetic sequence are 62 and 23 respectively.
(a) Find the general term of the arithmetic sequence.
(b) Find the 20th term and the 25th term of the arithmetic sequence.
17. Find the general term T(n) of the arithmetic sequence where the 5th term is 57 and the sum
of the 2nd term and the 3rd term is 59.
18. In an arithmetic sequence, the 1st term is 4 and the 4th term is
1
of the 5th term.
2
(a) Find the common difference of the arithmetic sequence.
(b) Find the 17th term of the arithmetic sequence.
19. Joan plans to save some money each Saturday so that the weekly balances of the savings
$13, $19, $25, $31,  form an arithmetic sequence. At least after how many weeks should
the balance of the savings be over $100?
20. There are a total of 12 stickers in red, blue and green in a bag. The number of stickers in
each colour in the bag forms an arithmetic sequence and the product of these numbers is 28.
If the majority are green stickers, find the number of green stickers.
Ex.14B Intermediate Set
 Intermediate Set
Level 1

21. For each of the following general terms T(n), find the 1st term and the common difference.
(a) T(n)  3n  9
(c) T(n)  2  7n
(b) T(n)  5(n  3)
3
(d) T (n)  9n 
4
Chapter 14 Arithmetic and Geometric Sequences
14.9
22. Find the general term T(n) of each of the following arithmetic sequences.
(a) 35, 26, 17, 8, 
1 1 3 5
(c)  , , , , 
8 8 8 8
(e) x  3, x  7, x  11, x  15, 
(b) 55, 44, 33, 22, 
1
1
5
4
(d) ,  ,  ,  , 
6
3
6
3
(f) a, 3a  2, 5a  4, 7a  6, 
23. Consider the arithmetic sequence 125, 112, 99, 86,  .
(a) Find the general term.
(b) Find the 21st term.
24. Consider the arithmetic sequence 7, 19, 31, 43,  .
(a) Find the general term.
(b) Find the 30th term.
(c) If the nth term is 391, find the value of n.
26. Insert two numbers between 31 and 17 such that these four numbers form an arithmetic
sequence.
27. Insert three numbers between 12 and 56 such that these five numbers form an arithmetic
sequence.
28. Find the number of terms in each of the following arithmetic sequences.
(a) 10, 19, 28, 37,  , 235
(c) 22, 15, 8, 1,  , 195
1
1
(e) 13, 11 , 10, 8 ,  , 20
2
2
(b) 5, 2, 1, 4,  , 70
2 1
(d) 2, 2 , 3 , 4,  , 12
3 3
(f) x  7, 4x  4, 7 x  1,  , 31x  23
29. The common difference of an arithmetic sequence is 5. The 15th term is 49. Find the
general term of the arithmetic sequence.
30. In the figure, p, q and r form an arithmetic sequence. If
p  73, find r.
B
p
q
O
r
A
C
Ex.14B Intermediate Set
25. If 8, a, b, c and 60 form an arithmetic sequence, find the values of a, b and c.
14.10
New Trend Mathematics S5A — Supplement
31. Consider the arithmetic sequence x  4, 2x  3, 30  x,  .
(a) Find the value of x.
(b) Find the general term of the arithmetic sequence.
32. 9  x, 13  3x, 24  6x,  is an arithmetic sequence.
(a) Find the value of x.
(b) Find the 14th term of the arithmetic sequence.
Level 2
33. The 6th and 11th terms of an arithmetic sequence are 31 and 66 respectively.
(a) Find the 1st term and the common difference of the arithmetic sequence.
(b) Find the general term of the arithmetic sequence.
34. The 4th and 9th terms of an arithmetic sequence are 11 and 34 respectively. Find the 11th
and 15th terms of the arithmetic sequence.
Ex.14B Intermediate Set
35. Find the general term T(n) of the arithmetic sequence where the 6th term is 104 and the
sum of the 3rd and 4th terms is 168.
36. In an arithmetic sequence, the 3rd term is 10 and the 4th term is
1
of the 8th term.
3
(a) Find the common difference of the arithmetic sequence.
(b) Find the 18th term of the arithmetic sequence.
37. From May 2005, Joseph’s monthly balances of a bank account $9 700, $9 050, $8 400,
$7 750,  form an arithmetic sequence. Which month would be the last month with a
positive balance? What was the amount of the balance?
38. There are a total of 18 apples, oranges and pears in a basket. The number of each type of
fruit in the basket forms an arithmetic sequence and the product of these numbers is 120. If
the majority are pears, find the number of pears.
39. (a) Find the first and the last multiples of 6 between 100 and 200.
(b) Find the number of integers between 100 and 200 which are divisible by 6.
40. For an arithmetic sequence, the product of the 2nd and 6th terms is 65, and the product of
the 1st and 7th terms is 45. Find the 1st term and the common difference of the arithmetic
sequence.
41. Consider the arithmetic sequence log a, log ab, log ab 2 ,  where a > 0 and b > 0.
(a) Find the common difference of the arithmetic sequence.
(b) Find T(9)  T(3).
Chapter 14 Arithmetic and Geometric Sequences
 Advanced Set
Level 1
14.11

42. Find the general term T(n) for each of the following arithmetic sequences.
(a) 43, 37, 31, 25, 
1 7 11 1
(c) , , , 1 , 
4 12 12 4
(e) 3  x, 7  2x, 11  3x, 15  4x, 
(b) 37, 24, 11, 2, 
2 8 5 2
(d) , , 1 , 2 , 
9 9 9 9
(f) a  2, 3a  3, 5a  8, 7a  13, 
43. Consider the arithmetic sequence 123, 109, 95, 81,  .
(a) Find the general term.
(b) Find the 42nd term.
44. Consider the arithmetic sequence 17, 30, 43, 56,  .
(a) Find the general term.
(b) Find the 35th term.
45. If 7, a, b, c, d and 73 form an arithmetic sequence, find the values of a, b, c and d.
46. Insert three numbers between 35 and 9 such that these five numbers form an arithmetic
sequence.
47. Insert four numbers between 25 and 72 such that these six numbers form an arithmetic
sequence.
48. Find the number of terms in each of the following arithmetic sequenc es.
(a) 6, 13, 20, 27,  , 181
(b) 84, 75, 66, 57,  , 6
(c) 17, 25, 33, 41,  , 233
(d)
2
1
(e) 19, 17 , 16 , 15,  ,  29
3
3
1
3 1
1
, 1, 1 , 2 ,  , 12
4
4 2
4
(f) 2x  7, 6x  10, 10 x  13,  , 50 x  43
49. The common difference of an arithmetic sequence is 7 and the 11th term is 59. Find the
general term of the arithmetic sequence.
50. The four interior angles of a quadrilateral form an arithmetic sequence. If the smallest
angle is 81, find the size of the largest angle.
Ex.14B Advanced Set
(c) If the nth term is 251, find the value of n.
14.12
New Trend Mathematics S5A — Supplement
51. Consider the arithmetic sequence 9x  2, 7x  1, 2x  14,  .
(a) Find the value of x.
(b) Find the general term of the arithmetic sequence.
52. 2x  9, 8x  11, 3x  57,  is an arithmetic sequence.
(a) Find the value of x.
(b) Find the 17th term of the arithmetic sequence.
Level 2
53. The 2nd and 5th terms of an arithmetic sequence are 27 and 15 respectively.
(a) Find the 1st term and the common difference of the arithmetic sequence.
(b) Find the general term of the arithmetic sequence.
54. The 3rd and 11th terms of an arithmetic sequence are 29 and 67 respectively. Find the
23rd and 31st terms of the arithmetic sequence.
Ex.14B Advanced Set
55. Find the general term T(n) of the arithmetic sequence where the 16th term is 41 and the
sum of the 3rd and 5th terms is 14.
56. In an arithmetic sequence, the 9th term is 16 and the 3rd term is
2
of the 6th term.
5
(a) Find the common difference of the arithmetic sequence.
(b) Find the 20th term of the arithmetic sequence.
57. For an arithmetic sequence, the sum of the 1st, 2nd and 3rd terms is 73 and the sum of the
4th, 5th and 6th terms is 127.
(a) Find the 1st term of the arithmetic sequence.
(b) Find the sum of the 7th, 8th and 9th terms.
58. Jimmy borrowed $15 000 from his mother on 15th January 2005 and he promised to repay
by instalments at the beginning of each month. If the monthly balances of the loan
$14 550, $14 100, $13 650, $13 200, , starting from the end of February 2005, form an
arithmetic sequence, which month would be the last month to have the loan fully repaid?
What was the amount of the last instalment?
59. In a mini theatre, there are 12 seats in the first row and the number of seats in each row
(starting from the 2nd row) is three more than that in the preceding row.
(a) Express the number of seats in the nth row in terms of n.
(b) If there are 54 seats in the last row, how many rows of seats are there in the mini
theatre?
Chapter 14 Arithmetic and Geometric Sequences
14.13
60. There are a total of 36 secondary 1, 2 and 3 students in a choir. The number of students
from each form in the choir forms an arithmetic sequence and the product of these numbers
is 528. If the majority of them are secondary 1 students, find the number of secondary 1
students in the choir.
61. (a) Find the first and the last multiples of 8 between 100 and 300.
62. For an arithmetic sequence, the product of the 3rd and 8th terms is 300, and the product of
the 5th and 6th terms is 396. Find the 1st term and the common difference of the arithmetic
sequence.
63. Consider an arithmetic sequence log a, log 10 a 3 , log 100 a 5 ,  where a > 0.
(a) Find the common difference of the arithmetic sequence.
(b) Find T(10)  T(5).
Ex.14B Advanced Set
(b) Find the number of integers between 100 and 300 which are divisible by 8.
64. In a graphic design, red, yellow and green squares are arranged in the following way:
(1) There is one red square in the 1st row, three red squares in the 2nd row, five red
squares in the 3rd row, and so on.
(2) In every alternate row starting from the 2nd row, there is one yellow square
between every two red squares.
(3) In every alternate row starting from the 3rd row, there is one green square between
every two red squares.
(a) Find the number of red squares in the 75th row.
(b) Find the number of yellow squares in the 78th row.
(c) Find the number of green squares in the 79th row.
Exercise 14C
 Elementary Set
Level 1

1. (a) 5  7  9  11    10th term
(b) 5  2  9  16    20th term
(c) 18  29  40  51    15th term
2. (a) 2  4  6    100
(b) 6  9  12  15    84
(c) 74  70  66  62    2
3. The general term of an arithmetic sequence is given by T(n)  3n  5. Find the sum of the
first 12th term.
Ex.14C Elementary Set
Find the sum of each of the following arithmetic series. (1  2)
14.14
New Trend Mathematics S5A — Supplement
4. Find the number of terms in each of the following arithmetic series whose sums are shown
below.
(a) 1  7  13  19    1 160
(b) 81  73  65  57    450
(c) 65  58  51  44    276
5. The 4th term of an arithmetic sequence is 92 and the common difference is 4. Find the sum
of the first 10 terms of the arithmetic sequence.
6. The 4th and 8th terms of an arithmetic sequence are 24 and 52 respectively.
(a) Find the 1st term and the common difference of the arithmetic sequence.
(b) Find the sum of the first 10 terms of the arithmetic sequence.
7. The sum of the first 15 terms of an arithmetic sequence is 735 and the 1st term is 21.
(a) Find the common difference.
(b) Find the 7th term.
Ex.14C Elementary Set
Level 2
8. (a) Find the sum of the arithmetic series 1  3  5  7    99.
(b) Hence find the product of a, a 3 , a 5 , a 7 ,  , a 99 .
9. Given the arithmetic sequence 9, 2, 5, 12,  ,
(a) find the 10th term.
(b) find the sum of the first 10 terms.
(c) find the sum of the 11th to 25th terms.
10. 612 bottles of wine are put in a way that the top layer has 17 bottles and each layer below
has 2 more bottles. How many layers are there?
11. The general term of an arithmetic sequence is given by T(n)  5  4n. Find the sum of the
series T(1)  T(3)  T(5)  T(7)    T(37).
12. (a) Find the smallest positive term of the arithmetic sequence 82, 75, 68,  .
(b) Find the sum of the arithmetic series 82  75  68    smallest positive term.
13. Find the sum of the positive terms of the arithmetic sequence 135, 121, 107, 93,  .
Chapter 14 Arithmetic and Geometric Sequences
14.15
15. The sums of the first 3 terms and the first 7 terms of an arithmetic sequence are 174 and
350 respectively.
(a) Find the general term of the arithmetic sequence.
(b) Find the smallest positive term of the arithmetic sequence.
16. (a) Find the smallest and largest multiples of 11 between 200 and 500.
(b) Find the sum of all multiples of 11 lying between 200 and 500.
Ex.14C Elementary Set
14. The general term of an arithmetic sequence is given by T(n)  4n  7. Find the value of n
such that the sum of the first n terms of the arithmetic sequence is 187.
17. (a) Find the sum of all multiples of 2 lying between 100 and 200 inclusive.
(b) Find the sum of all multiples of 4 lying between 100 and 200 inclusive.
(c) Between 100 and 200 inclusive, find the sum of all integers which are multiples of 2
but not multiples of 4.
 Intermediate Set
Level 1

Find the sum of each of the following arithmetic series. (18  19)
18. (a) 52  48  44  40    42nd term
(b) 22  ( 17)  (12)  (7)    36th term
19. (a) 7  15  23  31    135
(b) 8  17  26  35    161
(c) 82  71  60  49    (127)
20. The general term of an arithmetic sequence is given by T(n)  6  3n. Find the sum of the
first 15 terms.
21. Find the number of terms in each of the following arithmetic series whose sums are shown
below.
(a) 2  8  14  20    3 040
(b) 81  75  69  63    441
(c) 43  36  29  22    1 296
Ex.14C Intermediate Set
(c) 70  57  44  31    50th term
14.16
New Trend Mathematics S5A — Supplement
22. The 14th term of an arithmetic sequence is 76 and the common difference is 7. Find the
sum of the first 25 terms of the arithmetic sequence.
23. The 6th and 11th terms of an arithmetic sequence are 29 and 64 respectively.
(a) Find the 1st term and the common difference of the arithmetic sequence.
(b) Find the sum of the first 20 terms of the arithmetic sequence.
24. The sum of the first 13 terms of an arithmetic sequence is 871 and the 1st term is 19.
(a) Find the common difference.
(b) Find the 8th term.
Level 2
25. Find the product of x 2 , x 5 , x 8 , x 11 ,  , x 44 .
Ex.14C Intermediate Set
26. There are 592 seats in a cinema. The seats are arranged in a way that there are 7 seats in the
first row and 4 more seats in each succeeding row. How many rows are there in this
cinema?
27. The general term of an arithmetic sequence is given by T(n)  9  6n. Find the sum of the
following series.
T(2)  T(6)  T(10)    T(46)
28. (a) Find the smallest positive term of the arithmetic sequence 111, 97, 83, 69,  .
(b) Find the sum of the arithmetic series 111  97  83  69    smallest positive term.
29. Find the sum of the positive terms of the arithmetic sequence 371, 353, 335, 317,  .
30. The general term of an arithmetic sequence is given by T(n)  5n  6. Find the value of n
such that the sum of the first n terms of the arithmetic sequence is 1 599.
31. The sums of the first 3 terms and the first 12 terms of an arithmetic sequence are 339 and
978 respectively.
(a) Find the general term of the arithmetic sequence.
(b) Find the smallest positive term of the arithmetic sequence.
Chapter 14 Arithmetic and Geometric Sequences
14.17
32. The sums of the first 4 terms and the first 6 terms of an arithmetic sequence are 6 and 39
respectively.
(a) Find the 1st term and the common difference.
(b) Find the sum of the first 10 terms.
(c) Find the sum of the 11th to 20th terms.
33. (a) Find the smallest and largest multiples of 7 between 200 and 500.
(b) Find the sum of all multiples of 7 lying between 200 and 500.
34. (a) Find the sum of all multiples of 3 lying between 200 and 400.
(b) Find the sum of all multiples of 9 lying between 200 and 400.
(c) Between 200 and 400, find the sum of all integers that are multiples of 3 but not
multiples of 9.
36. There are 10 pumps in a water tank. It takes a worker 5 minutes to start up one pump. The
first pump starts working at 9:00 p.m., the second at 9:05 p.m., the third at 9:10 p.m. and so
on. Each pump pumps out 50 L of water in one minute. All pumps stop working at
10:30 p.m.
(a) How long (in minute) has each of the first, second and third pumps been working for
from 9:00 p.m. to 10:30 p.m.?
(b) How long (in minute) has the last pump been working for from 9:00 p.m. to
10:30 p.m.?
(c) Find the volume of water pumped out by the pumps from 9:00 p.m. to 10:30 p.m.
37. The office hours of a company are from 9:00 a.m. to 5:00 p.m. Monday to Friday. Due to
the heavy workload, the company requires its employees to work overtime next month. The
following are two overtime options for its employees to choose from.
Option 1: Work extended office hours from 8:00 a.m. to 6:00 p.m. every working day.
Option 2: Work normal office hours in the first working day of the month. Then, work
15 minutes more every subsequent working day.
It is given that there are 22 working days in the next month.
(a) Find the total overtime next month under each option.
(b) Employees working more overtime will receive bonus wages. Which option should be
chosen if an employee wants to earn more?
(c) After how many days will the total overtime under option 1 be equal to that under
option 2?
Ex.14C Intermediate Set
35. Find the general term of the sequence 2  5, 2  5  8, 2  5  8  11, 2  5  8  11  14,  .
14.18
New Trend Mathematics S5A — Supplement
 Advanced Set
Level 1

Find the sum of each of the following arithmetic series. (38  39)
38. (a) 5  (9)  (13)  (17)    30th term
(b) (2  3 )  (7  3 )  (12  3 )  (17  3 )    35 th term
x x 3x
(c)
 
 x    37 th term
4 2 4
39. (a) 7  10.5  14  17.5    98
(b) 5  1  3  7    59
(c) 7 5  5  5 5  11 5    47 5
40. The general term of an arithmetic sequence is given by T(n)  9  7n. Find the sum of the
first 30 terms.
41. Find the number of terms in each of the following arithmetic series whose su ms are shown
below.
Ex14C Advanced Set
(a) 3  10  17  24    780
(b) 82  74  66  58    350
(c) 39  30  21  12    1 221
42. The 15th term of an arithmetic sequence is 78 and the common difference is 7. Find the
sum of the first 20 terms of the arithmetic sequence.
43. The 4th and 12th terms of an arithmetic sequence are 43 and 75 respectively.
(a) Find the 1st term and the common difference of the arithmetic sequence.
(b) Find the sum of the first 13 terms of the arithmetic sequence.
44. The sum of the first 17 terms of an arithmetic sequence is 969 and the 1st term is 21.
(a) Find the common difference.
(b) Find the 10th term.
Level 2
45. Find the product of x 4 , x 8 y3 , x 12 y 6 , x 16 y 9 ,  , x 100 y 72 .
46. Billy repays a loan of $30 000 (including interest) under a special plan. At the end of the
first month, he will repay $5 000. At the end of each of the following months, he will repay
$500 more than the previous month until the loan is fully repaid. How long does it take him
to repay the loan?
Chapter 14 Arithmetic and Geometric Sequences
14.19
47. The general term of an arithmetic sequence is given by T(n)  11  7n. Find the sum of the
following series.
T(11)  T(14)  T(17)    T(74)
48. Find the sum of the arithmetic series 124  103  82  61    smallest positive term.
49. Find the sum of the negative terms of the arithmetic sequence 291, 278, 265, 252,  .
50. The general term of an arithmetic sequence is given by T(n)  8n  13. Find the value of
n such that the sum of the first n terms of the arithmetic sequence is 1 575.
51. The sums of the first 4 terms and the first 20 terms of an arithmetic sequence are 338 and
1 030 respectively.
(a) Find the general term of the arithmetic sequence.
52. The sum of the first 20 terms of an arithmetic sequence is 400 and that of the first 16 terms
of the arithmetic sequence is 448. Find the sum of the 15th to 25th terms.
53. Find the sum of all multiples of 13 lying between 200 and 700.
54. (a) Find the sum of all multiples of 4 lying between 400 and 600 inclusive.
(b) Find the sum of all multiples of 6 lying between 400 and 600 inclusive.
(c) Between 400 and 600 inclusive, find the sum of all integers that are multiples of 4 but
not multiples of 6.
55. Find the general term of the sequence
7  10, 7  10  13, 7  10  13  16, 7  10  13  16  19,  .
56. (a) Find the sum of the first 15 terms of the arithmetic sequence a, 3a  b, 5a  2b, 7a  3b,  .
Express your answer in terms of a and b.
(b) Hence find the sum of the first 15 terms of the arithmetic sequence 100, 302, 504, 706,  .
57. Sarah has a keep-fit plan. On the 1st of October, she will jog 12 minutes. For each of the
following days, she will jog 3 minutes more than the previous day until she jogs 30 minutes
a day. Then she will keep on jogging for 30 minutes a day. Assume that she goes jogging
every day,
Ex.14C Advanced Set
(b) Find the largest negative term of the arithmetic sequence.
14.20
New Trend Mathematics S5A — Supplement
Ex.14C Advanced Set
(a) on which day will she start jogging for 30 minutes a day?
(b) If 9 units of fat can be burnt for every minute of jogging, find the amount of fat burnt
by jogging for Sarah in October.
58. There are two options for Carl to repay a loan of $x (including interest) in 12 months.
Option 1: Repay $k each month.
Option 2: Repay $a in the first month. In each of the following months, repay $800 more
than the previous month.
(a) Express k in terms of a.
(b) Is 3 000 a reasonable value of x? Explain briefly.
Exercise 14D
 Elementary Set
Level 1

1. For each of the following general terms T(n), find the first term and the common ratio.
1 1
(a) 3(5) n1
(b)  (  ) n 1
3 2
2
(c) 5(4) n
(d) 4( ) n
3
2. Find the common ratio and the general term of each of the following geometric sequences.
(a) 2, 12, 72, 432, 
Ex.14D Elementary Set
(c) 64, 96, 144, 216, 
(b) 3, 12, 48, 192, 
1
(d) 4, 2, 1,  , 
2
1
3. The 2nd term of a geometric sequence is 64 and its common ratio is . Find the 5th term
2
of the geometric sequence.
4. The 1st term of a geometric sequence is 4 and its common ratio is 3.
(a) Find the general term of the geometric sequence.
(b) If T(n)  972, find the value of n.
2
5. The 3rd term of a geometric sequence is 8 and its common ratio is . Find the 1st term of
3
the geometric sequence.
6. The 6th term of a geometric sequence is 48 and the 10th term is 3. Find the common ratio
of the geometric sequence.
Chapter 14 Arithmetic and Geometric Sequences
14.21
1 1 1 1
,
, , , .
81 27 9 3
(a) Find the general term of the geometric sequence.
7. Consider the geometric sequence
(b) Find the 10th term of the geometric sequence.
8. Find the common ratio and the number of terms in each of the following geometric
sequences.
(a) 6, 12, 24, 48,  , 1 536
(c) 128, 192, 288, 432,  , 2 187
(b) 15, 45, 135, 405,  , 3 645
32 64
512
(d) 24, 16,
,
, ,
3 9
243
Level 2
9. It is given that T(1)  3, T(2)  6, T(3)  12 and T(4)  24 are the first 4 terms of a geometric
sequence. Find the common ratio and the number of terms of the sequence T(1), T(3), T(5),
T(7),  , 12 288 .
(a) Find the 9th term of the geometric sequence.
(b) If the nth term is 8 times the 9th term, find the value of n.
11. There is 2 cm 3 of air in a balloon at present. Air is pumped into the balloon such that the
volume of air in the balloon increases by 20% for each pump. The balloon will burst if the
volume of air inside is greater than 2 700 cm 3. After how many pumps will the balloon
burst?
12. The 3rd and 8th terms of a geometric sequence are 72 and 2 304 respectively. Find the 1st
term and the common ratio of the geometric sequence.
13. The 3rd term of a geometric sequence is 81 and the 7th term is 625. Find the general
term(s) of the geometric sequence.
14. Find the general term T(n) of a geometric sequence if
T (5)
14
 8 and T (5)  T (2) 
.
T (2)
3
15. In a geometric sequence, the sum of the 1st and 2nd terms is 120, and the sum of the 2nd
and 3rd terms is 80.
(a) Find the common ratio of the geometric sequence.
(b) Find the 1st term of the geometric sequence.
Ex.14D Elementary Set
10. Consider the geometric sequence 36, 18, 9,  .
14.22
New Trend Mathematics S5A — Supplement
16. Insert one number between 162 and 18 such that these three numbers form a geometric
sequence.
Ex.14D Elementary Set
17. Insert two numbers between 6 and 48 such that these four numbers form a geometric
sequence.
18. If x, x  6, 4x  24,  is a geometric sequence, find the value of x.
19. Consider four numbers 16, x, y, 48.
(a) Given that the first three numbers form a geometric sequence, express y in terms of x.
(b) Given that the last three numbers form an arithmetic sequence, express y in terms of x.
(c) Hence find the values of x and y.
 Intermediate Set
Level 1

20. If the general term of a geometric sequence is T(n)  4(2)n , find the 1st term and the
common ratio of the geometric sequence.
Ex.14D Intermediate Set
21. Find the common ratio and the general term of each of the following geometric sequences.
128
(a) 5, 10, 20, 40, 
(b) 144, 96, 64,
,
3
3
3
(c) 2, 6, 18, 54, 
(d) 6, 3, ,  , 
2
4
22. The 1st term of a geometric sequence is 729 and its common ratio is
1
.
3
(a) Find the general term of the geometric sequence.
(b) Find the 6th term of the geometric sequence.
1
1
23. The 1st term of a geometric sequence is 64 and its common ratio is . If T (n) 
, find
2
32
the value of n.
24. The 4th term of a geometric sequence is 256 and its common ratio is 2. Find the general
term of the geometric sequence.
25. The 1st term of a geometric sequence is 288 and the 4th term is
ratio of the geometric sequence.
243
. Find the common
2
Chapter 14 Arithmetic and Geometric Sequences
14.23
26. Find the common ratio and the number of terms in each of the following geometric
sequences.
(a) 3, 9, 27, 81,  , 6 561
2
(c) 162, 54, 18, 6,  ,
81
(b) 64, 96, 144, 216,  , 729
8 16 32
256
(d) 4, , ,
,,
3 9 27
729
Level 2
7
27. Consider the geometric sequence 28, 14, 7, ,  . If the 12th term is 8 times the nth term,
2
find the value of n.
28. The first term and the common ratio of a geometric sequence are 6 and 4 respectively. If
the nth term is smaller than 10 000, find the greatest value of n.
1 024
30. The 2nd and 7th terms of a geometric sequence are 32 and
respectively. Find the 1st
243
term and the common ratio of the geometric sequence.
31. Find the general term(s) T(n) of a geometric sequence if T(5)  T(3)  36 and
T (5) 1
 .
T (3) 4
32. In a geometric sequence, the sum of the 2nd and 5th terms is 72, and the sum of the 5th and
8th terms is 9.
(a) Find the common ratio of the geometric sequence.
(b) Find the 1st term of the geometric sequence.
33. Insert two numbers between 81 and 192 such that these four numbers form a geometric
sequence.
34. Insert three numbers between 7 and 112 such that these five numbers form a geometric
sequence.
35. Consider a geometric sequence x  3, 2x  1, 6x  3,  .
(a) Find the value of x.
(b) Find the general term of the geometric sequence.
Ex.14D Intermediate Set
29. Since 1 January 2004, Mr. Lam has been investing 10% of his monthly salary in a fund
each month for the education of his daughter. Suppose his monthly salary was $ 25 000 in
2004 and it increases by 2% each year, in which year will the money invested be greater
than $35 000?
14.24
New Trend Mathematics S5A — Supplement
36. Consider four numbers 48, x, y, 144.
(a) Given that the first three numbers form a geometric sequence, express y in terms of x.
(b) Given that the last three numbers form an arithmetic sequence, express y in terms of x.
(c) Hence find the values of x and y.
Ex.14D Intermediate Set
37. Three numbers form an arithmetic sequence. The 1st term minus the 3rd term is 6. When
the 1st, 2nd and 3rd terms increase by 7, 4 and 3 respectively, the resulting numbers form a
geometric sequence.
(a) Find the common difference of the arithmetic sequence.
(b) Find these three numbers.
38. The perimeter of A1B1C1 is 192 cm. The
mid-points of its sides are joined to form a smaller
triangle A2 B2C 2 and the process goes on (see the
figure).
(a) (i) Find the ratio of the perimeter of
A 1B 1C 1 to that of A2 B2C 2 .
(ii) Express the perimeter of An BnC n in
terms of n.
(b) Find the perimeter of A6 B6C 6 .
 Advanced Set
Level 1
A1
A3
C2
B3
B1
B2
C3
A2
C1

39. The general term of a geometric sequence is given by T(n)  22n . Find the common ratio of
the geometric sequence.
Ex.14D Advanced Set
40. Find the general term of each of the following geometric sequences.
(a) 4, 12, 36, 108, 
1 1
1
1
,
(c)  , ,  ,
4 16
64 256
256
(b) 288, 192, 128, 
,
3
4 2
3
(d) , , 1, , 
9 3
2
41. The 1st term of a geometric sequence is 4 and its common ratio is 3. If T(n)  2 916, find
the value of n.
42. The 5th term of a geometric sequence is 4 608 and its common ratio is 
T ( n)  
2 187
, find the value of n.
2
3
. If
4
Chapter 14 Arithmetic and Geometric Sequences
14.25
43. Find the common ratio and the number of terms in each of the following geometric
sequences.
(a) 10, 20, 40,  , 20 480
512
(c) 1 458, 972, 648,  ,
9
(b) 3, 12, 48,  , 49 152
325
40 625
(d) 26, 65,
,,
2
16
Level 2
find the value of n.
2 4 8
,
, ,  . If the nth term is 216 times the 10th term,
81 27 9
45. The first term and the common ratio of a geometric sequence are 12 and
the nth term is greater than 15 000, find the least value of n.
5
respectively. If
3
46. Mr. Li’s monthly salary in 2004 was $25 000 and he saved 4% of it each month. Mr. Chan’s
monthly salary in 2004 was $18 000 and he saved 5% of it each month. If Mr. Li’s and Mr.
Chan’s monthly salaries increase by 5% and 8% each year respectively, in which year will
the monthly savings of Mr. Chan be greater than Mr. Li’s?
9
9
and
respectively. If all the terms
2
32
of the geometric sequence are positive, find the 1st term and the common ratio of the
geometric sequence.
47. The 4th and 8th terms of a geometric sequence are
48. Find the general term(s) T(n) of a geometric sequence if T(5)  T(3)  12 and
T (5) 2
 .
T (3) 3
49. In a geometric sequence, the sum of the 5th and 6th terms is 240, and the sum of the 3rd
and 4th terms is 60.
(a) Find the common ratio of the geometric sequence.
(b) Find the 1st term of the geometric sequence.
50. Insert two numbers between 320 and 5 such that these four numbers form a geometric
sequence.
51. Insert three numbers between 12 and 432 such that these five numbers form a geometric
sequence.
52. If x, 2x  12, 3x  18,  is a geometric sequence, find the value of x.
Ex.14D Advanced Set
44. Consider the geometric sequence
14.26
New Trend Mathematics S5A — Supplement
53. Consider the geometric sequence x  8, x  10, 4x  5,  .
(a) Find the value(s) of x.
(b) Find the general term(s) of the geometric sequence.
54. Consider four numbers 36, x, y, 75.
(a) Given that the first three numbers form an arithmetic sequence, express y in terms of x.
(b) Given that the last three numbers form a geometric sequence, express y2 in terms of x.
(c) Hence find the values of x and y.
55. Three numbers form an arithmetic sequence. The sum of the 1st and 3rd terms is 10. When
the 1st, 2nd and 3rd terms increase by 20, 13 and 9 respectively, the resulting numbers form
a geometric sequence.
(a) Find the 2nd term.
(b) Find the 1st and 3rd terms.
Ex.14D Advanced Set
56. Three numbers x, y and z are in the ratio 1 : 3 : 5. If 2 and 4 are added to x and z respectively,
they form a geometric sequence.
(a) Express the geometric sequence in terms of x.
(b) Find x, y and z.
57. Three numbers x, y and z are in the ratio 2 : 3 : 6. If 12, 18 and 9 are added to x, y and z
respectively, they form a geometric sequence. Find x, y and z.
58. In the figure, ABCD is a square with sides of 18 cm
each. The four points A 1 , B 1 , C 1 and D 1 divide the sides
AB, BC, CD and DA respectively in the ratio 1 : 2, and
they are joined to form another square A 1 B1 C 1 D 1 .
Squares A2 B2C 2 D2 , A3 B3C 3 D3 ,  are then formed in
the same pattern.
(a) Find the length of each of the following sides.
(i) A1B1
(ii) A2 B2
(iii) A3 B3
(Leave your answers in surd form if necessary.)
A1
B
A
A2
A3
D3
B1
D2
B2
B3
C
(b) (i) Find the ratio of the area of A1B1C1D1 to that of A2 B2C 2 D2 .
(ii) Express the area of An BnC n Dn in terms of n.
(c) Find the area of A10 B10C10 D10 .
(Correct your answer to 3 significant figures.)
2 500
(d) If the area of An Bn C n Dn is
cm2 , find the value of n.
81
C3
C2
C1
18 cm
D1
D
Chapter 14 Arithmetic and Geometric Sequences
Exercise 14E
 Elementary Set
Level 1
14.27

1. Find the sum of each of the following geometric series.
(a) 5  10  20  40    8th term
(b) 2  8  32  128    9th term
(c) 64  96  144  216    7th term
(d) 6 3  18  18 3  54    12th term
(Leave your answers in surd form if necessary.)
2. Find the 1st term of each of the following geometric sequences.
(a) Common ratio  2, sum of the first 10 terms  341.
(b) Common ratio  3, sum of the first 6 terms  1 456.
2
(c) Common ratio  , sum of the first 5 terms  422.
3
3. For each of the following geometric sequences, find the number of terms n and the sum of
the first n terms S(n).
(b) 256, 384, 576, 864,  , 6 561
(c) 2 048, 1 024, 512,  , 8
4. Find the sum of the 2nd to 12th terms of the geometric sequence 7, 21, 63, 189,  .
5. The 1st term of a geometric sequence is 6 and the 4th term is
2
.
9
(a) Find the common ratio of the geometric sequence.
(b) Hence find the sum of the first 6 terms.
6. The first 3 terms of a geometric sequence are x, x  4 and x  10.
(a) Find the value of x.
(b) Find the sum of the first 6 terms.
7. T(n) is the general term of the geometric sequence 2 2 , 4, 4 2 , 8,  .
(a) Find the common ratio and the number of terms of the geometric sequence T(1), T(3),
T(5),  , T(15).
(b) Find the sum of T(1)  T(3)  T(5)    T(15).
8. The 1st term of a geometric sequence is 8 and the sum of the first 3 terms is 14.
(a) If the common ratio of the geometric sequence is r, show that 4r2  4r  3  0.
(b) Hence find the 2nd term.
Ex.14E Elementary Set
(a) 2, 6, 18,  , 4 374
14.28
New Trend Mathematics S5A — Supplement
Level 2
9. The common ratio of a geometric sequence is
1
, the nth term is 6 and the sum of the first
3
n terms is 240.
(a) Find the 1st term of the geometric sequence.
(b) Find the value of n.
10. (a) Find the sum of the first n terms of the geometric sequence 1, 3, 9, 27,  in terms of n.
(b) If the sum of the first n terms of the geometric sequence in (a) is greater than 5 000,
find the least value of n.
Ex.14E Elementary Set
11. There are 8 prizes, the 1st prize, 2nd prize,  , and 8th prize, in a competition. The value
of the 8th prize is $35. The value of the 7th prize is triple that of the 8th prize. The value of
the 6th prize is triple that of the 7th prize and so on. Find the total value of the 8 prizes.
12. Sammy invests $10 000 in a fund each year with a return of 4% p.a.
(a) Find the value of the investment held by Sammy in the fund at the end of the 3rd year,
correct your answer to the nearest dollar.
(b) At least after how many years will the value of the investment held by Sammy in the
fund exceed $1 000 000?
13. A bank offers an interest of 5% p.a. compounded yearly on an equal sum of money
deposited at the beginning of each year. Let $x be the sum deposited each year.
(a) Find, in terms of x, the total amount accumulated
(i) at the end of the 1st year.
(ii) at the end of the 2nd year.
(b) Show that the total amount accumulated at the end of the nth year is $21(1.05 n  1)x.
(c) If Jane deposits $30 000 in the bank at the beginning of each year, find the total amount
accumulated at the end of the 8th year, correct your answer to the nearest dollar.
Ex.14E Intermediate Set
 Intermediate Set
Level 1

14. Find the sum of each of the following geometric series.
3
(a)  3  12  48    8th term
(b) 64  32  16  8    9th term
4
36
 6    10 th term
(c) 36 6  36 
6
(Leave your answers in surd form if necessary.)
Chapter 14 Arithmetic and Geometric Sequences
14.29
15. Find the 1st term of each of the following geometric sequences.
(a) Common ratio  2, sum of the first 10 terms  3 069.
(b) Common ratio  3, sum of the first 6 terms  3 640.
3
(c) Common ratio   , sum of the first 4 terms  525.
4
16. For each of the following geometric sequences, find the number of terms n and the sum of
the first n terms S(n).
1
(a) 6, 18, 54, 162,  , 4 374
(b) 6 561, 2 187, 729, 243,  , 
3
(c) 3, 12, 48, 192, , 12 288
17. Find the sum of the 3rd to 11th terms of the geometric sequence 3, 12, 48, 192,  .
18. The 1st term of a geometric sequence is 3 and the 4th term is 
3
.
8
(b) Hence find the sum of the first 8 terms.
19. The first 3 terms of a geometric sequence are x  4, x  4, x  20.
(a) Find the value of x.
(b) Find the sum of the first 10 terms.
20. T(n) is the general term of the geometric sequence 2, 2 3, 6, 6 3,  .
(a) Find the common ratio and the number of terms of the geometric sequence T(2), T(4),
T(6),  , T(18).
(b) Find the sum of T(2)  T(4)  T(6)    T(18).
21. The 1st term of a geometric sequence is 4 and the sum of the first 3 terms is 39.
(a) If the common ratio of the geometric sequence is r, show that 4r 2  4r  35  0.
(b) Hence find the 3rd term.
Level 2
2
22. The common ratio of a geometric sequence is , the nth term is 24 and the sum of the first
5
n terms is 609.
(a) Find the 1st term of the geometric sequence.
(b) Find the value of n.
Ex.14E Intermediate Set
(a) Find the common ratio of the geometric sequence.
14.30
New Trend Mathematics S5A — Supplement
23. (a) Find the sum of the first n terms of a geometric sequence 8, 24, 72, 216,  in terms of
n.
(b) If the sum of the first n terms of the geometric sequence in (a) is greater than 2 000,
find the least value of n.
24. The interest rate of a bank is 6% p.a. compounded yearly. Ivy plans to deposit $15 000 at
the beginning of each year.
(a) Find the amount obtained at the end of the 10th year, correct your answer to the nearest
dollar.
Ex.14E Intermediate Set
(b) At least after how many years will Ivy obtain an amount of $300 000?
25. The interest rate of a bank is 3% p.a. compounded yearly. Rita plans to deposit a fixed sum
in the bank each year in order to get an amount of $400 000 after eight years. How much is
the fixed sum? (Correct your answer to 1 decimal place.)
26. The sum of the first 6 terms of a geometric sequence is 65 times the sum of the first
3 terms.
(a) Find the common ratio of the geometric sequence.
(b) Find the 1st term if the 4th term is greater than the 2nd term by 180.
27. Consider a sequence with T(1)  2, T(2)  11, T(3)  101, T(4)  1 001.
(a) Find T(n).
(b) Hence find the sum of T(1)  T(2)  T(3)    T(n) in terms of n.
 Advanced Set
Level 1

Ex.14E Advanced Set
28. Find the sum of each of the following geometric series.
25
125
(a) 12  (10)  ( )  (
)    6 th term
3
18
3
 1  3  3    11 th term
(b)
3
(Leave your answers in surd form if necessary.)
29. Find the 1st term of each of the geometric sequences.
(a) Common ratio  2, sum of the first 8 terms  3 060.
3
(b) Common ratio   , sum of the first 7 terms  2 653.
4
(c) Common ratio 
5 , sum of the first 10 terms  781( 5  1) .
Chapter 14 Arithmetic and Geometric Sequences
14.31
30. For each of the following geometric sequences, find the number of terms n and the sum of
the first n terms S(n).
65 536
(a) 27, 36, 48, 64,  ,
243
(b) 6, 6 5, 30, 30 5,  , 3 750
(Leave your answers in surd form if necessary.)
31. Find the sum of the 5th term to the 9th term of the geometric sequence 3, 12, 48, 192,  .
27
2
and the 5th term is .
3
8
(a) Find the common ratio of the geometric sequence.
32. The 1st term of a geometric sequence is
(b) Hence find the sum of the first 8 terms.
(a) Find the values of x.
(b) Find the sum of the first 7 terms of each geometric sequence.
34. T(n) is the general term of the geometric sequence 3 3, 9, 9 3, 27,  .
(a) Find the common ratio and the number of terms of the geometric sequence T(3), T(6),
T(9),  , T(24).
(b) Find the sum of T(3)  T(6)  T(9)    T(24).
(Leave your answers in surd form if necessary.)
35. The 1st term of a geometric sequence is 12 and the sum of the first 3 terms is 57.
(a) If the common ratio of the geometric sequence is r, show that 4r 2  4r  15  0.
(b) Hence find the 4th term.
Level 2
36. The common ratio of a geometric sequence is 
first n terms is 133.
(a) Find the 1st term of the geometric sequence.
(b) Find the value of n.
2
, the nth term is 32 and the sum of the
3
37. (a) Find the sum of the first n terms of the geometric sequence 108, 72, 48, 32,  in terms
of n.
(b) If the sum of the first n terms of the geometric sequence in (a) is greater than 300, find
the least value of n.
Ex.14E Advanced Set
33. The first 3 terms of a geometric sequence are 2x  8, 2x  28, x  2.
14.32
New Trend Mathematics S5A — Supplement
38. The increase of the height of a tree drops by 5% each year. Last year, it was 5 m tall and
now it is 5.5 m tall.
(a) Find the height of the tree after 10 years. (Correct your answer to 1 decimal place.)
(b) At least after how many years will the tree be taller than 14 m?
39. In the figure, isosceles right-angled triangles
A1B1C1, A2 B2C 2 , A3 B3C3 ,  are drawn such that
A1
B1
B2
A2
C1
A3
B3

Ex.14E Advanced Set
the lengths of B1C1 , B2C 2 , B3C 3 ,  form a
BC
geometric sequence and 2 2  1.6 . If there are
B1C1
10 triangles in the figure and the total area of these
triangles is 7 750 cm 2, find B1C1 , correct your
answer to 3 significant figures.
C2
C3
97
40. The sum of the first 8 terms of a geometric sequence is
times the sum of the first
16
4 terms.
(a) Find the common ratio of the geometric sequence.
(b) Find the 1st term if the 4th term is greater than the 3rd term by 45.
41. Consider a sequence with T(1)  1, T(2)  11, T(3)  111, T(4)  1 111.
(a) Find T(n).
(b) Hence find the sum of T(1)  T(2)  T(3)  T(4)    T(n) in terms of n.
Exercise 14F
[ In this exercise, leave your answers in surd form if necessary. ]
Ex.14F Elementary Set
 Elementary Set
Level 1

Find the sum to infinity of each of the following geometric sequence s. (1  2)
3
1. (a) 108, 36, 12, 4, 
(b) 12, 6, 3, , 
2
2
2
1
1
1
1
(c) 10, 2,  ,  , 
(d)  ,  ,  , 
,
5 15
45 105
5
25
Chapter 14 Arithmetic and Geometric Sequences
2. (a) 16, 8, 4, 2, 
(c) 50, 40, 32, 25.6, 
14.33
8
(b) 9, 6, 4,  , 
3
2 1 1 1
(d)
, ,
, ,
2 2 2 2 4
3. Find the sum of each of the following infinite geometric series.
(a) 22  2.2  0.22  0.022  
(b) 200  120  72  43.2  
4. Write down the 1st term of the geometric sequence satisfying the given conditions.
1
3
(a) r  , S()  1
(b) r  , S()  6
3
4
1
1
(c) r   , S()  12
(d) r   , S()  20
4
3
(a) a  5, S()  6
(b) a  8, S()  20
(c) a  12, S()  9
(d) a  20, S()  14
6. Write down the first 3 terms of the geometric sequence satisfying the given conditions.
3
2
(a) r  , S()  20
(b) r   , S()  30
3
4
(c) a  4, S()  28
(d) a  12, S()  10
7. Express the following recurring decimals as fractions.
(a) 0.5
(b) 0.2 5
(c) 0.25
8. The 2nd and 3rd terms of a geometric sequence are 48 and 32 respectively.
(a) Find the common ratio of the geometric sequence.
(b) Find the 1st term of the geometric sequence.
(c) Find the sum to infinity of the geometric sequence.
Level 2
2k
form a geometric sequence.
3
(a) Find the value of k.
(b) Find the sum to infinity of the geometric sequence.
9. It is given that 4, k,
Ex.14F Elementary Set
5. Write down the common ratio of the geometric sequence satisfying the given conditions.
14.34
New Trend Mathematics S5A — Supplement
10. T(1), T(2), T(3),  , T(n),  is a geometric sequence.
(a) If T(2)  216 and T(5)  8, find T(1) and the common ratio of the geometric sequence.
(b) Show that T(2), T(4), T(6), T(8),  is a geometric sequence.
(c) Hence find the sum of T(2)  T(4)  T(6)  T(8)   .
11. The sum to infinity of a geometric sequence T(1), T(2), T(3), T(4),  is 36 and the sum of
T(1)  T(3)  T(5)  T(7)   is 27.
(a) Find the common ratio.
(b) Find T(1).
Ex.14F Elementary Set
12. An oil well produced 2.5 million barrels of oil in the first year. Thereafter, the annual
production of the oil well decreases to 90% of that in the preceding year.
(a) How many years does it take for this oil well to produce more than 20 million barrels
of oil?
(b) Show that no matter how long the oil well operates, its total oil production will not
exceed 25 million barrels.
13. The figure is formed by an infinite number of arcs of semi-circles with diameters AB, BC,
1
1
CD,  , where BC  AB, CD  BC,  , and so on. If AB  8 cm, find the perimeter of
2
2
the figure, express your answer in terms of .
A
C
E
D
B
Ex.14F Intermediate Set
 Intermediate Set
Level 1

14. Find the sum to infinity of each of the following geometric sequence s.
8 8
2
(a) 24, 8, , , 
(b) 18, 6, 2,  , 
3
3 9
64
(c) 25, 20, 16,
(d) 625, 125 5, 125, 25 5, 
,
5
Chapter 14 Arithmetic and Geometric Sequences
14.35
15. Find the sum of each of the following infinite geometric series.
(a) 888  88.8  8.88  0.888  
(b) 100  40  16 
32

5
16. Write down the 1st term of the geometric sequence satisfying the given conditions.
1
3
(a) r  , S()  4
(b) r  , S()  12
3
4
1
2
(c) r   , S()  16
(d) r   , S()  28
5
3
(a) a  4, S()  12
(b) a  6, S()  10
(c) a  14, S()  8
(d) a  15, S()  9
18. Write down the first 3 terms of the geometric sequence satisfying the given conditions.
3
2
(a) r  , S()  20
(b) r   , S()  28
7
5
(c) a  10, S()  6
(d) a  30, S()  25
19. Express the following recurring decimals as fractions.
(a) 0.7
(b) 0.2 7
(c) 0.27
20. The 3rd and 4th terms of a geometric sequence are 72 and 45 respectively.
(a) Find the common ratio of the geometric sequence.
(b) Find the 1st term of the geometric sequence.
(c) Find the sum to infinity of the geometric sequence.
Level 2
3k
form a geometric sequence.
4
(a) Find the value of k.
21. It is given that 8, k,
(b) Find the sum to infinity of the geometric sequence.
22. T(1), T(2), T(3),  , T(n),  is a geometric sequence.
(a) If T(3)  216 and T(6)  64, find T(1) and the common ratio of the geometric sequence.
(b) Show that T(1), T(3), T(5), T(7),  is a geometric sequence.
(c) Hence find the sum of T(1)  T(3)  T(5)  T(7)   .
Ex.14F Intermediate Set
17. Write down the common ratio of the geometric sequence satisfying the given conditions.
14.36
New Trend Mathematics S5A — Supplement
23. The sum to infinity of a geometric sequence T(1), T(2), T(3), T(4),  is 75 and the sum of
T(1)  T(3)  T(5)  T(7)   is 45.
(a) Find the common ratio.
(b) Find T(1).
24. When a hot air balloon starts leaving the
ground, it rises 80 m from the ground in the first
minute. Thereafter, the distance risen per minute
is 20% of that in the preceding minute.
(a) Find the distance of the balloon from the
ground 3 minutes after it has been started
leaving the ground.
(b) What is the maximum distance of the
balloon from the ground?
Ex.14F Intermediate Set
25. In the figure, an infinite number of semi-circles,
with diameters AB, BC, CD, , are drawn on a
straight line. It is given that the lengths of the A
diameters form a geometric sequence, where
AB  18 cm and BC  15 cm.
(a) Find the sum of the lengths of all the diameters.

B
C
D

  
(b) Find the sum to infinity of AB  BC  CD   .
(c) Hence find the perimeter of the figure.
(Express your answer in terms of .)
26. Brian and Sarah are racing against each other on a straight road. Brian and Sarah start at A
and B respectively, which are 48 m apart. Each of them is running at a uniform speed, but
the speed of Brian is 1.5 times that of Sarah. When Brian reaches B, Sarah will be at C.
When Brian reaches C, Sarah will be at D, and so on.
...
A
48 m
B
C
D
E
(a) Find the respective distances of BC, CD and DE.
(b) Show that AB, BC, CD,  form a geometric sequence. Hence find the common ratio.
(c) How far from A does Brian have to run before he overtakes Sarah?
Chapter 14 Arithmetic and Geometric Sequences
 Advanced Set
Level 1
14.37

27. Find the sum to infinity of each of the following geometric sequences.
9
27
1
(a) 8, 6, ,  , 
(b) 12 3, 6,  3, , 
2
8
2
28. Find the sum of each of the following infinite geometric series.
(a) 111  11.1  1.11  0.111  
(b) 160  120  90 
135

2
29. Write down the 1st term of the geometric sequence satisfying the given conditions.
2
3
(a) r  , S()  54
(b) r  , S()  18
3
5
2
4
(c) r   , S()  42
(d) r   , S()  22
5
7
(a) a  15, S()  20
(b) a  12, S()  30
(c) a  18, S()  10
(d) a  35, S()  25
31. Write down the first 3 terms of the geometric sequence satisfying the given conditions.
3
4
(a) r  , S()  45
(b) r   , S()  21
7
5
(c) a  15, S()  9
(d) a  22, S()  12
32. Express the following recurring decimals as fractions.
(a) 0.2 4
(b) 0.2 04
(c) 0.24
(d) 0.20 4
33. The 5th and 6th terms of a geometric sequence are 25 and 5 5 respectively.
(a) Find the common ratio of the geometric sequence.
(b) Find the 1st term of the geometric sequence.
(c) Find the sum to infinity of the geometric sequence.
Level 2
34. The 3rd and 7th terms of a geometric sequence are 36 and 4 respectively.
(a) Find the 1st term and common ratio of the geometric sequence.
(b) Find the sum to infinity of the geometric sequence.
Ex.14F Advanced Set
30. Write down the common ratio of the geometric sequence satisfying the given conditions.
14.38
New Trend Mathematics S5A — Supplement
3k
form a geometric sequence.
5
(a) Find the value of k.
(b) Find the sum to infinity of the geometric sequence.
35. It is given that 9, k,
36. T(1), T(2), T(3),  , T(n),  is a geometric sequence.
(a) If T(3)  324 and T(6)  96, find T(1) and the common ratio of the geometric sequence.
(b) Show that T(1), T(4), T(7), T(10),  is a geometric sequence.
(c) Hence find the sum of T(1)  T(4)  T(7)  T(10)   .
37. The sum to infinity of a geometric sequence T(1), T(2), T(3), T(4),  is 96 and the sum of
T(2)  T(4)  T(6)  T(8)   is 72.
(a) Find the common ratio.
(b) Find T(1).
Ex.14F Advanced Set
38. The sum of the first two terms of a geometric sequence is 9 and the sum to infinity is 25.
(a) Find the common ratio.
(b) Write down the first 3 terms of the sequence.
39. In the figure, A1B1C1D1 is a square with sides of
15 cm and A2 B2C 2 D2 is another square inscribed
in A1B1C1D1 such that A 2 , B 2 , C 2 and D 2 divide
A1 B1 , B1C1 , C1 D1 and D1 A1 respectively in the
ratio 3 : 2. Following this pattern, squares are
drawn successively.
B1
D3
D4
C4
D2
A3
C3
(a) Find the length of A2 B2 .
(b) If the sides A1 B1 , A2 B2 , A3 B3 ,  of the
squares form a geometric sequence, find the
common ratio.
A2
A1
A4
B2
B4
B3
D1
C2
C1
(c) (i) Find the sum of the perimeters of the infinite number of squares drawn.
(ii) If an insect crawls along A1 A2 , A2 A3 , A3 A4 ,  , find the total distance travelled
by the insect before it reaches the centre of the squares.
(d) If the areas of the squares A1B1C1D1 , A2 B2C 2 D2 ,  , An BnC n Dn ,  also form a
geometric sequence, find the sum of the areas of the infinite number of squares drawn .
Chapter 14 Arithmetic and Geometric Sequences
40. In the figure, B1C1CD is a square inscribed in the
right-angled triangle ABC where C  90, BC  a,
4a
AC 
and B1C1  b .
3
(a) Express b in terms of a.
14.39
A
B4
C
D4 4
B3
C3
D3
B2
D2
(b) B2C 2C1D1 is a square inscribed in  AB1C1 .
Express B2C 2 in terms of a.
B1
D1
C2
C1
(c) When squares B3C 3C 2 D2 , B4C 4C 3 D3 ,  are
drawn successively, B1C1 , B2C 2 , B3C 3 , 
form a geometric sequence.
B
C
D
(i) Write down the length of B4C 4 in terms of
a.
(ii) Find, in terms of a, the sum of the areas of
the infinite number of squares drawn in
this pattern.
...
A
B
C
D E
54 m
In order to find the distance travelled by the woman before she overtakes the girl, consider
the following:
Assume the woman starts at A and the girl starts at B. (See the figure)
When the woman reaches B, the girl will be at C.
When the woman reaches C, the girl will be at D, and so on.
The distances between the woman and the girl, AB, BC, CD, DE,  , will diminish.
Thus, the required distance is the sum to infinity of the sequence AB, BC, CD, DE, .
(a) Write down the first three terms of the sequence.
(b) Show that AB, BC, CD,  form a geometric sequence. Hence find the common ratio.
(c) Find the required distance.
42. In the figure, there is a square with sides of x1 each. A right-angled triangle with sides x 1,
x 2 and y 1 is formed on one side of the square where x 1 : x 2 : y 1  5 : 4 : 3. Another square
with sides x 2 each is formed on one side of the right-angled triangle. Then, another
right-angled triangle with sides x 2, x 3 and y 2 is formed on one side of this square where
x 2 : x 3 : y 2  5 : 4 : 3. The process goes on infinitely.
Ex.14F Advanced Set
41. On a straight road, a woman is 54 m behind a girl. It is known that each of them walks at a
uniform speed but the woman walks 2.5 times as fast as the girl.
14.40
New Trend Mathematics S5A — Supplement
y3
y2
x3
x3
x2
x4
x4
y4
Ex.14F Advanced Set
x2
y1
x1
(a) Show that x1, x2, x3,  form a geometric sequence and find the common ratio.
(b) Show that x2 y 1, x3 y2, x4 y3,  form a geometric sequence and find the common ratio.
(c) If x 1  10 cm,
(i) find the sum of the areas of all the squares in the figure.
(ii) find the area of the whole figure.
CHAPTER TEST
(Time allowed: 1 hour)
Section A
1. Consider the arithmetic sequence 8, 3, 2, 7,  .
(a) Find the general term.
(b) Find the 12th term.
(c) Find the sum of the first 12 terms of the sequence.
2 2 2
2. Consider the geometric sequence 6, , ,
, .
3 27 243
(a) Find the common ratio of the sequence.
(b) Find the sum to infinity of the sequence.
(1 mark)
(1 mark)
(2 marks)
(1 mark)
(2 marks)
3. Find the number of terms in the arithmetic series 33  26  19   whose sum is 312.
(3 marks)
4. Find the sum of the geometric series 7  14  28    3 584.
(4 marks)
5. The 2nd term of an arithmetic sequence is 43 and the sum of the first 16 terms of the
sequence is 40.
(a) Find the general term of the sequence.
(4 marks)
(b) Find the smallest positive term of the sequence.
(2 marks)
Chapter 14 Arithmetic and Geometric Sequences
14.41
6. It is given that 4k, k, 3 form a geometric sequence where k  0.
(a) Find the value of k.
(2 marks)
(b) Find the sum to infinity of the geometric sequence 4k, k, 3,  .
(3 marks)
Section B
7. In a park, a path is paved with rectangular tiles in the same width but different lengths. The
tiles are arranged from the smallest to the largest, and their lengths form an arithmetic
sequence. The length of the smallest tile is 0.6 m, the length of the largest tile is 1.8 m, and
the difference in length between any adjacent tiles is 0.05 m.
0.6 m
..........
..........
..........
..........
1.8 m
the same width
the same width
(a) (i) Find the number of tiles used.
(ii) Hence find the length of the path.
(4 marks)
(b) Suppose the 13th tile is a square,
(i) find the width of the path.
(ii) Hence find the area of the path.
(iii) If the area of the nth tile is 1.14 m 2, find the value of n.
(6 marks)
8. a and b are positive numbers. a, 12, b form a geometric sequence and 12, b, a form an
arithmetic sequence.
(a) Find the value of ab.
(2 marks)
(b) Find the values of a and b.
(3 marks)
(c) (i) Find the sum to infinity of the geometric sequence a, 12, b,  .
(ii) Find the sum to infinity of all the terms that are negative in the geometric sequence
a, 12, b,  .
(5 marks)
Multiple Choice Questions [ 3 marks each ]
9. If the common difference of the
arithmetic sequence x 1, x 2, x 3,  is d, the
common difference of the arithmetic
sequence 2 x1  3, 2 x2  3, 2 x3  3,  is
10. How many positive integers less than
150 are not divisible by 11?
A. 135
B. 136
A. 2d.
C. 137
B. d  3.
D. 138
C. 2d  3.
D. 2(d  3).
□
□
14.42
New Trend Mathematics S5A — Supplement
11. Which of the following could be an
arithmetic sequence/arithmetic sequences?
1 1
1
1
I.
,
,
,
,
2 22 222 2 222
II. 0.2, 0.22, 0.222, 0.222 2, 
III. log 2, log 0.2, log 0.02, log 0.002, 
15. If a, b, c, d form a geometric sequence,
which of the following sets of numbers
form geometric sequences?
I. ab, bc, cd
II. a  b, b  c, c  d
a b c
III. , ,
b c d
A. I only
A. I and II only
B. II only
B. I and III only
C. III only
C. II and III only
D. II and III only
□
12. If the sum of the first n terms of an
arithmetic sequence is 4n 2  3n, find the
5th term.
D. I, II and III
16. The general term of a sequence is given
by T ( n)  2 n  2 . Find the product of the
first 10 terms.
A. 39
A. 2 35
B. 47
B. 2 36
C. 76
C. 2 53
D. 115
□
13. If x2, x2, 3x3,  is a geometric sequence
and x  0, find the value of x.
A. 3
B. 3
1
C. 
3
1
D.
3
□
B. ab
D.
□
17. The 3rd and 5th terms of a geometric
sequence are 60 and 15 respectively.
Find the sum to infinity of the geometric
sequence if all the terms of the sequence
are positive.
B. 240
A. 1
ab
1
ab
D. 2 55
A. 160
14. If a, x1, x2 , b and a, y1 , y2 , y3 , b are
two geometric sequences where a and b
are positive numbers, find the value of
x1x2
.
y1 y2 y3
C.
□
□
C. 320
D. 480
□
18. If the sum of the first 3 terms of a
19
geometric sequence is equal to
of the
27
sum to infinity of this sequence, find the
common ratio.
2
A. 
3
2
B.
3
3
C. 
2
3
D.
2
□