Download Q.1 A progression has a first term 12 and a fifth term of 18.

Sequences
Topics
a. Sequence or progression: It is a collection of terms, in a definite order, obeying a certain
rule.
i. 2, 8, 32, 128, …
iii. 5, 2, -1, -4, -7, -11, …
ii. 3, 7, 11, 15, 19, 23,…
iv. 2, 10, 50, 250,...
b. Notation of a sequence: The terms of a sequences are denoted by 𝑎1, 𝑎2 , 𝑎3 , … are the
first, second and third terms of the sequence.
c. Arithmetic Sequence or progression (AP): If a sequence is obtained by adding a constant
number to previous term to obtain next term. The constant number is called common
difference and it is denoted by 𝑑. The first term is denoted by 𝑎or𝑎1 .
d. Formula of AP i.e. 𝑎𝑛 = 𝑎 + (𝑛 − 1 )𝑑
e. Application of formula.
i.
To find the specific term of an AP
ii.
To find the value of 𝑎and 𝑑 when value of two terms are given.
iii.
To find the number of term when progression is finite.
EXERCISE 8C
Q.1
In each of the following AP’s, two terms are given; find the common ratio and first term
i. 4th term =15, 9th term = 35
ii. 2nd term = 2x, 11th term = -7x
iii. 3rd term = 2p +7, 7th term = 4p +19
Q.2
Find the number of term of each of the following AP’s
i. 5, 7, 9, …, 111
iii. 1.8, 1.2, 0.6, …, -34.2
ii. -14, -10, -6, …, 94
Topics
𝑛
a. To find the sum of first “𝑛” terms of an AP. 𝑠𝑛 = 2 [2𝑎 + (𝑛 − 1)𝑑].
b. Application of sum formula.
i. To find the sum of said numbers of terms.
ii. To find other missing unknown.
iii. Combination of terms
EXERCISE 8C and Miscellaneous Exercise
Q.3
Find how many terms are given arithmetic series must be taken to reach the given sum.
i.
Q.4
Q.5
Q.6
20 +23+ 26+… sum=680
ii.
-11 -4 +3 +… sum =2338
The 𝑟𝑡ℎ term of an AP is1 + 4𝑟. Find, in term of 𝑛, the sum of the first 𝑛 terms of the
progression.
The sum of first two terms of an AP is 18 and the sum of first four terms is 52.Find the
sum of first eight terms.
The sum of first twenty terms of an AP is 50, and the sum of next twenty terms is 50.Find the sum of first hundred terms of the progression.
Compiled By : Sir Rashid Qureshi
www.levels.org.pk
1
An Arithmetic progression has first term 𝑎and the common difference -1.The sum of
first n terms is equal to sum of first 3n terms. Express 𝑎in terms of 𝑛.
Q.7
Topics.
a. Geometric Sequence or Progression (GP): If a sequence is obtained by multiplying a
constant number to previous term to obtain next term. The constant number is called
common ratio and it is denoted by 𝑟. The first term is denoted by 𝑎 or𝑎1 .
b. Formula for GP:𝑎𝑛 = 𝑎𝑟 𝑛−1 and its applications
i. To find the specific term of an GP
ii. To find the value of 𝑎 and common ratio when value of two terms are
given.
iii. To find the number of term when progression if finite.
Q.8
In each of the following GP’s, two terms are given; find the common ratio and first term
i. 4th term =19683,
9th term = 81
ii. 3rd term = 8,
9th term = 64
iii. 3rd term =6 ,
7th term = 96
Q.9
Find the number of term of each of the following GP’s
i. 2, 4, 8, …, 2048
ii. 16, 12, 9 …, 3.796875
iii. 5, -10, 20, … -40960
Topics
a. To find the sum of first “n” terms of an AG. 𝑠𝑛 =
b.
c.
i.
ii.
iii.
Q.10
Q.11
Q.12
Q.13
𝑎
𝑎(1−𝑟 𝑛 )
1−𝑟
.
Sum to infinity of geometric series. 𝑠∞ = 1−𝑟.
Application of sum formula.
To find the sum of said numbers of terms.
To find other missing unknown.
Combination of terms.
Find the sum, for the given number of terms, of each of the following geometric series.
Give the answer correct to 4d.p.
i.
2 +6 +18 +… ,
10 terms
4
ii.
12 – 4 + 3 … ,
10 terms
iii.
3 +6 + 12 +… ,
12 terms
Fund the sum of each of the following Geometric series.
i. 1 +2 +4 +…, 1024
1 1
1
ii. 1 - 3 + 9 - …, -19683
if 𝑥, 𝑦 and 𝑧 are the first three terms of a geometric sequence, show that𝑥 2 , 𝑦 2 and 𝑧 2
form another geometric sequence.
Different numbers 𝑥, 𝑦, 𝑧 are the first three terms of the GP with common ratio r and
also the first , second and forth terms of an AP.
a. Find the value of r,
b. For which term of AP will next be equal to term of geometric progression.
Compiled By : Sir Rashid Qureshi
www.levels.org.pk
2
Q.14
Q.15
Find sum to infinity of the following geometric series
1 1
iii. 10 -5 +2.5 + …
i. 1 + 2 + 4 + …,
ii. 0.1 +.001 +.001 + …
1
A geometric series has first term 𝑎 and the common ratio .Show that the sum to
√2
infinity of the series is (2 + √2) .
Q.16 A geometric series has non zero first term 𝑎 and common ratio r. Given that the sum of
the first 8 terms of the series is equal to the half the sum to infinity, find the value of r.
Given that 17th term of the series is 10. Find the value of r.
Past papers Questions
Compiled by: Rashid Ali Qureshi
Q.1
A progression has a first term 12 and a fifth term of 18.
i. Find the sum of first 25 terms if the progression is arithmetic.
ii. Find the 13th term if the progression is geometric.
Q.2
A geometric progression has first term 64 and sum to infinity 256. Find
i. The common ratio
ii. The sum of first 19 terms.
Q.3
Each a year a company gives grant to a charity. The amount given each year increased
by 5% of its value in the preceding year. The grant in 2001 was $5000. Find
i. The grant given in 2011.
ii. The total amount of money given to the charity during the year 2001 to 2011 inclusive.
(J2006)
Q.4
The first term of an AP s 8 and the common difference is 𝑑where𝑑 ≠ 0. The first term,
the fifth term and eighth term of this AP are the first term, the second term and the
third term, respectively, of a GP whose common ratio is 𝑟.
3
i. Write down two equations connecting 𝑑 𝑎𝑛𝑑 𝑟. Hence show that 𝑟 = 4 and find the
value of 𝑑.
ii. Find the sum to infinity of GP.
iii. Find the sum of first eight terms of AP.
(N2009)
Q.5
A progression has second term of 96 and fourth term of 54. Find the first term of the
progression in each of the following cases:
i. The progression is geometric.
ii. The progression is geometric with positive common ratio.
(N2009)
2
Q.6
i.
Find the sum to infinity of th GP with first three terms 0.5, 0.5 and 0.53.
ii.
The firste two terms in an AP is 5 and 9. The last term in the progression is only
term which is greater than 200. Find the sum of all terms in the progression.
(J2009)
Q.7
The ninth term of an AP is 22 and the sum of first four terms is 49.
i. Find the first term of the progression and the common difference.
The 𝑛𝑡ℎ term of the progression is 46.
ii. Find the value of 𝑛.
(J2010)
Q.8
Find the sum of multiples of 5 between 100 and 300 inclusive.
2
A geometric progression has a common ration of − 3 and sum of first three terms is 35.
Find
Compiled By : Sir Rashid Qureshi
www.levels.org.pk
3
i.
ii.
Q.9
Q.10
Q.11
The first term of the progression.
The sum to infinity.
(J2010)
th
The first term of GP is 12 and the second term is −6. Find the 10 term of the
progression and sum to infinity.
(J2010)
The first term of an AP is 6 and te fifth term is 12. The progression has 𝑛 terms and the
sum of all the terms is 90. Find the value of 𝑛.
(N2008)
The first term of GP is 81 and fourth term is 24. Find
i. The common ratio of the progression.
ii. The sum to ininity of the progression.
iii. The second term and third terms of this geometric progression are the first and
fourth terms respectively of an AP.Find the sum of first ten turms of this AP.
(J2008)
Q.12
i.
The second term of GP is 3 and sum to infinity is 12. Find the first term of the
progression.
ii. An AP has same first term and second term as the geometric progression.Find the
sumof firste 20 terms of the AP.
(J2007)
Q.13 The first term of an AP is 𝑎 and common difference is 𝑑, where𝑑 ≠ 0.
i. Write down an expression in terms 𝑎 and 𝑑, for the 5th and 15th terms.
ii. The first term, the 5th term and the 15th term of AP are the firste three terms of a
geometric progression. Show that 3𝑎 = 8𝑑, Hence find the value of 𝑟.
(N2007)
Q.14
i. Find the sum of all integers between 100 and 400 that are divisible by 7.
ii. The first term in GP are 144, 𝑥 and 64 respectiveely, where 𝑥is positive. Find the value of
𝑥 and sum to infinity of prigression.
(N2006)
Q.15 A geometric progression has 6 terms. The first term is 192 and the common ratio is 1.5.
An arithmetic progression has 21 terms and common difference 1.5. Given that the sum
of all the terms in the geometric progression is equal to the sum of all the terms in the
arithmetic progression, find the first term and the last term of the arithmetic
progression.
(J2005)
Q.16 A small trading company made a profit of $250 000 in the year 2000. The company
considered two different plans, plan A and plan B, for increasing its profits. Under plan
A, the annual profit would increase each year by 5% of its value in the preceding year.
Find, for plan A,
(i) the profit for the year 2008,
(ii) the total profit for the 10 years 2000 to 2009 inclusive.
Under plan B, the annual profit would increase each year by a constant amount $D.
(iii) Find the value of D for which the total profit for the 10 years 2000 to 2009 inclusive
would be the same for both plans.
(N2005)
Q.17 Find
(i) the sum of the first ten terms of the geometric progression 81, 54, 36. . .
(ii) the sum of all the terms in the arithmetic progression 180, 175, 170. . . 25.
(N2004)
Compiled By : Sir Rashid Qureshi
www.levels.org.pk
4
Q.18
(a) A debt of $3726 is repaid by weekly payments which are in arithmetic progression.
The firstpayment is $60 and the debt is fully repaid after 48 weeks. Find the third
payment.
(b) Find the sum to infinity of the geometric progression whose first term is 6 and whose
second term is 4.
(N2003)
Q.19 In an arithmetic progression, the 1st term is −10, the 15th term is 11 and the last term is
41. Find the sum of all the terms in the progression.
(J2003)
Q.20 A progression has a first term of 12 and a fifth term of 18.
(i) Find the sum of the first 25 terms if the progression is arithmetic.
(ii) Find the 13th term if the progression is geometric.
(J2002)
Answer:
1.
a. 3, 4
12. 6, -450
b. 3𝑥, −𝑥
13. 2.5
1
14. 8140, 71034
c. 𝑃 + 1, 2 𝑝 + 3
15. 205
2.
16. 369000, 3140000, 14300
a. 54
17. 239, 320
b. 61
18. 61.5, 18
c. 28
19. 542.5
3.
20. 750, 40.5
a. 20
4.
5.
6.
7.
8.
b. 38
𝑛(2𝑛 + 3)
168
-750
1
2𝑛 − 2
,
1
a. 3, , 531441
b. ±√2, 4
1
c. ±2, 1 2
9. 11, 6, 14
10. 59048,8.998
29525
11. 2047, 39366
13. 3, 14th
1 20
14. 2, 9, 3
1
15. 6 4
16. 0.917, 40
Past Papers Questions:
10. n = 8
2
11. 3, 243, 270
Compiled By : Sir Rashid Qureshi
www.levels.org.pk
5