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Series and Sequences
Convergent Series
Arithmetic and Geometric Means
Series:
This is the sum of the terms in a sequence, 3+6+12... (It is as if the terms are
separated by a + or a – sign.
In an arithmetic series there are two different types of equations that can be used
depending upon which terms are known.
The first equation may be used if you already know the first and last terms of the
sequence and you only wish to find their total sum.
S (n) = n [t (1) + t (n)]
2
(sum) (#) (1st) (last
(of) (term) (term)
(terms)
Example: Find the sum of the first four terms in this series.
3, 5, 7, 9
1, 2, 3, 4
S (4) = 4(3+9)
2
S (4) = 24  The sum of the 4 terms.
The second equation may be used when the last term in unknown.
S (n) = n [2 t (1) + (n-1) d]
2
(sum) (#) (1st)
(#) (diff)
(of) (term) (your)
(terms) (looking)
(for)
Example: Find the sum of the first twenty terms in this series.
3, 5, 7, 9...
1, 2, 3, 4
S (20) = 20 [2(3) + (20-1) 2]
2
S (20) = 440  The sum of all the first 20 terms is 440.
In a geometric series only one equation is used to find the sum of a certain
number of terms in a sequence.
n  (# of terms)
S (n) = t (1) 1-r
1-r  (common ratio)
st
(sum) (1 )
(term)
Example: Find the sum of the first ten terms in this series.
25, 50, 100 ratio=2
10
S (10) = 25 1-2
1-2
S (10) = 25 1-1024
-1
S (10) = 25 (1023)
S (10) = 25575  The sum of all the first 10 terms of this sequence is 25575.
Sequences:
There are two types of sequences, arithmetic and geometric. A sequence is a
series of numbers, which have something in common with one another. In a sequence,
there are term values and term numbers. The term values are the numbers, which are
part of the sequence, which have something in common with one another. The term
numbers are just the numbers in the order of each term.
t (n) = term value
n = term number
Example:
t (n) = 3, 5, 7, 9, 11
n = 1, 2, 3, 4, 5
In this example t (n), each term value, is equal to double its term number plus one.
t (n) = 2n + 1
Does this work out?
t (2) = 2 (2) + 1
t (2) = 5
This equation does work out for this sequence.
An arithmetic sequence is a sequence in which one term equals a constant added
to the preceding term.
Example:
3, 5, 7, 9, 11
The constant would be two because two is being added on each time. In an
arithmetic sequence this is also called the common difference.
A way to determine if in fact it is an arithmetic sequence or not is to subtract term
one from term two and this should be equal to subtracting term two from term three.
t2 – t1 = t3 – t2
To find the value or a specific term number in this type of sequence this equation
may be used.
t (n) = t (1) + (n-1) d  (common difference)
(last) (1st) (term #)
(term) (term)(2 B found)
Example: What is the value of the 10th term.
t (n) = 3, 5, 7, 9, 11
n = 1, 2, 3, 4, 5
t (10) = 3 + (10-1) 2
t (10) = 21
In this case the 10th term would have a value of 21.
A geometric sequence is a sequence in which one term equals a constant,
multiplied by the previous term.
Example:
t (n) = 3, 6, 12, 24
n = 1, 2, 3, 4
The constant of this equation is two because two multiplies each number each
time. In a geometric sequence this is also called the common ratio.
A way to determine if it is a geometric sequence is to divide the second term by
the first term and see if it is equal to if you divide the third term by the second term.
t2 = t3
t1 t2
Example:
6 = 12
3 6
2=2
A way to find the value of a specific term number is to use this equation.
T (n) = t (1) (r
(last) (1st)
(term) (term)
n-1  (term #)
)  (common ratio
Example: What is the value of the 10th term.
10-1
t (10) = 3 (2
)
t (10) = 1536  The value of the 10th term is 1536
Convergent Series:
A series converges to a number five, if the sum (S (n)) stays arbitrarily close to
five as h gets very large.
Example:
15, -4.5, -1.35
First v much be found, to do this divide the terms to discover if they are equal.
t2 = t3 = -4.5 = -1.35
t1 t2
15
-4.5
-0.3 = -0.3
Once v is determined then this equation may be used to find the point in the
sequence where the numbers will stop increasing and will eventually remain constant.
S = t (1) = 15 = 11.54
1-v 1-(-0.3)
An example of a problem where convergent geometric series equations can be
used is.
If a nail gets hammered into a board 20 mm the first time and 18 mm more the
second time it forms a geometric sequence. What number is it going to converge upon
(what is the maximum distance that it will move).
Five v = 18 = 0.9
20
S = t (1) = 20 = 200 mm  This would be the total length of the nail.
1-v 1-0,9
Means:
Between two numbers are numbers that form arithmetic and geometric sequences
with the given numbers.
Example: Find four arithmetic mean between 37 and 54.
First of all four spaces must be places in between 37 and 54.
37, , , , , 54
The arithmetic the equation:
t (n) = t (1) + (n-1) d  Must be used to determine d or the common difference.
54 = 37 (6-1) d  The # six is used because it is the total # of terms in the sequence.
54-37 = (5) d
17 = 5 d
5
5
3.4 = d
Once d has been determined, the rest of the sequence may be completed. Adding
the common difference, 3.4 to each term in the sequence can do this.
37(+3.4), 40.4(+3.4), 43.8(+3.4), 50.6(+3.4), 54
The sequence has been completed.
For a geometric sequence the same type of methods are used only using the
geometric equation.
Example: Find three geometric means between 3 and 48.
3, , , , 48
The geometric equation:
n-1
t (n) = t (1) (r
)
5-1
48 = 3 (r
)
4
48 = r
16
_
4/16 = r
2=r
In this case the common ratio or “r” is +2 or – 2. This happens frequently with
geometric means because both +2 and –2 will work out correctly.
(+2) 3(2), 6(2), 12(2), 24(2), 48
 Both of these answers are correct.
(-2) 3(-2), -6 (-2), 12(-2), -24(-2), 48
Websites
Here are some helpful websites, to assist you in your learning.
1. http://www.sparknotes.com/math/precalc/sequencesandseries/
2. http://www.calculus.net/
3. http://www.alltel.net/~okrebs/page131.html
Practice Questions
Determine whether the sequence can be arithmetic or geometric, or if it is neither.
If it is arithmetic, find the common difference, if it is geometric, find the common ratio.
1. 7, 12, 17
2. 3, 6, 12
3. –1, 0, 1
4. 25, 50, 100
5. 2, -4, 8
Find the specific terms of the indicated arithmetic sequences:
Remember t (n) = t (1) + (n-1) d
1. 45th term of 2, 5, 8
2. 29th term of 7, 11, 15
3. 51st term of 18, 14, 10
4. 68th term of 95, 92, 87
Find the specified terms of the indicated geometric sequences:
n-1
Remember t (n) = t (1) (r
)
1. 7th term of 2, 6, 18
2. 10th term of 1, -2, 4
3. 51st term of the sequence for which t (1) = 7 and r = 1.02
4. 64th term of the sequence for which t (1) = 1 and r = -2
Arithmetic and Geometric means:
1. Find 4 arithmetic means between 37 and 54
2. Find 5 arithmetic means between –91 and –67
3. Find 3 geometric means between 81 and 16
4. Find 4 geometric means between 1/32 and 32
Arithmetic and Geometric series remember:
For arithmetic: S (n) = n ( t (1) + t (n) ) * When first and last terms are known.
2
For geometric: S (n) (2 t (1) + (n-1) d ) * When last term is unknown.
1. S (10) for 4 + 7 + 10
2.
S (13) for 9 + 20 + 31
3.
S (20) for the series with t (1) = 15 and d = 10
4.
S (30) for the series with t (1) = 17 and d = 8
Geometric series:
Remember
n
S(n) = t (1) 1-r
1-r
1. S (5) for 1 + 2 + 4 + …
2.
S (6) for 2 + 6 + 18+ …
3.
S (10) for the series with t (1) = 5 and r = 3
4.
S (7) for the series with t (1) = 6 and r = -2
Quiz/Test
1. Determine whether the sequence is arithmetic geometric or neither, state the common
difference or ratio.
a) 25, 50, 100
b) ½, ¼, 1/8
c) 3, 6, 12
d) –1, 0, 1
2. Find the specified terms for the following arithmetic sequences.
a) 51st term of 18, 14, 10
b) t (95) for 136, 131, 126
3. Find the specified terms of the indicated geometric sequence.
a) 8th term of 54, 18, 6
b) 37th term of the sequence for which t (1) = 1000 and r = 0.92
4. Find the indicated arithmetic means.
a) 3 between –257 and –397
b) 4 between 123 and 55
5. Find the indicated geometric means.
a) 2 between 5 and 135
b) 3 between 5 and 45
6. Find the specified terms in the indicated geometric series.
a) S (6) for 1, -3, 9
b) S (20) for the series with t (1) = 13 and r = 1.1
c) S (30) for the series with t (1) = 11 and r = 1.3
7. Find the specified terms in the indicated arithmetic series.
a) S (48) for the series with t (1) = 29 and d = -3
b) S (40) for the series with t (1) = 8 and t (6) = 38
c) S (50) for the series with t (1) = 7 and t (9) = 47
Appendix
Practice Questions, Answers:
1. Arithmetic - Common difference = 5
2. Geometric – Common ratio = 2
3. Arithmetic – Common difference = 1
4. Geometric – Common ratio = 2
5. Geometric – Common ratio = -2
1. t (45) = 2 + (45-1) 3
t (45) = 134
common difference = 3
2. t (29) = 7 + (29-1) 4
t (29) = 119
common difference = 4
3. t (51) = 18 + (51-1) –4
t (51) = -182
common difference = -4
4. t (68) = 95 + (68-1) –3
t (68) = -106
common difference = -3
1. t (7) = 2 (3
t (7) = 1458
r=3
7-1
)
10-1
2. t (10) = -1 (-2
)
t (10) = 512
r = -2
51-1
3. t (51) = 7 (1.02
)
t (51) = 18, 8411
r = 1.02
64-1
4. t (64) = 1 (-2
)
t (64) = -9.2233
r = -2
1. 37, 40.4, 43.8, 47.2, 50.6, 54
54 = 37 + (6-1) d
17 = 5 d
3.4 = d
2. –91, -87, -83, -79, -75, -71, -67
-67 = -91 + (7-1) d
24 = 6 d
4=d
3. 81, 53.946, 35.928, 23.928, 16
5-1
16 = 81 (r
)
4
0.19753 = r
_
4/0.19753 = r
4.1/32, 4/32, 16/32, 64/32, 8, 32
6-1
32 = 1/32 ( r
)
5
1024 = r
__
5/1024 = r
4=r
1. S (10) = 10 ( 4 + 10)
2
S (10) = 5 (14)
S (10) = 70
2. S (13) = 13 (9+31)
2
S (13) = 6.5 (40)
S (13) = 260
3. S (20) = 20 (2 (15) = (20-1) 10)
2
S (20) = 20 (30 + 190)
2
S (20) = 1200
4. S (30) = 30 (2 (17) + (30-1) 8)
2
S (30) = 30 (34 + 232)
2
S (30) = 3990
5
1. S (5) = 1 (1-2 ) = -31
1-2
1
S (5) = 31
6
2. S (6) = 2 (1-3 )
1-3
S (6) = -729
10
3. S (10) = 5 (1-3 )
1-3
S (10) = -147622.5
9
4. S (9) = 6 (1 + 2 )
1+2
S (9) = 1026
Quiz/Test, Answers:
1.
a) Geometric, common ratio = 2
b) Geometric, common ratio = 0.5
c) Geometric, common ratio = 2
d) Arithmetic, common difference = 1
2. a) t (51) = -182
d = -4
b) t (95) = -334
d = -5
3. a) t (8) = 118098
r=3
b) t (37) = 49.70
r = 0.92
4. a) d = -35
b) d = -13.6
5. a) r = 3
b) r = 1.732
6. a) S (6) = -182.5
b) S (30) = 2137.2
c) S (20) = 6931.466
7. a) S (48) = -1992
b) S (40) = 920
c) S (50) = 1350
Bibliography
1. Pre- Calculus Mathematics Two. Ontario: Oxford University Press, 1993.
2. Foerster A. Paul. Algebra and Trigonometry. Canada: Addison-Wesley
Publishing company, Inc, 1984.