Infinite Geometric Series Worksheet

Infinite Geometric Series Worksheet
  • Page 1
 1.  
Find the sum of the infinite geometric series.
55+1+55+5+555+5+....
a.
1.25
b.
does not exist
c.
2.25
d.
3.25


Solution:

r = a2a1=55+555+1
[Common ratio.]

= 55+5×5+15

= 55(5+1)×5+15=15
[Simplify.]

|r| = |15| < 1, so the series converges and sum exists.

S = a11-r
[Sum of infinite terms of geometric series.]

= 55+11-15

= 55+1×55-1 = 55-1 = 1.25
[Simplify.]

So, the sum of the infinite geometric series is 1.25.


Correct answer : (1)
 2.  
What is the sum of the infinite geometric series 5 + 2 + 0.8 + ...?
a.
25 7
b.
7.8
c.
does not exist
d.
25 3


Solution:

r = a2a1 = 2 / 5 = 0.4
[Replace a1 = 5 and a2 = 2.]

| r | = | 0.4 | = 0.4 < 1, so the series converges and the sum exists.

S = a11 - r = 5 / 1-0.4
[Use the formula and replace a1 = 5, r = 0.4.]

S = 5 / 0.6 = 25 / 3
[Simplify.]

So, the sum of the infinite geometric series is 25 / 3.


Correct answer : (4)
 3.  
The sum of an infinite geometric series is 252 and the common ratio is 1 7. Find the first term.
a.
288
b.
216
c.
7
d.
36


Solution:

S = a11-r
[Sum of an infinite geometric series.]

252 = a11-17
[Replace S = 252, r = 1 / 7.]

252(1 - 1 / 7) = a1
[Cross multiply.]

252 - 36 = a1
[Simplify.]

216 = a1

So, the first term of an infinite geometric series is 216.


Correct answer : (2)
 4.  
Find the sum of the infinite geometric series.
48 + 480 + 4800 + ......
a.
does not exist
b.
16 3
c.
5328
d.
48000


Solution:

r = a2a1 = 480 / 48 = 10
[Replace a1 = 48 and a2 = 480.]

Since | r | = | 10 | = 10 > 1, the series diverges, so the sum does not exist.


Correct answer : (1)
 5.  
Find the sum of the infinite geometric series.
- 196 - 28 - 4 - ......
a.
- 686 3
b.
Does not exist
c.
- 228
d.
- 168


Solution:

r = a2a1 = - 28- 196 = 1 / 7
[Replace a2 = - 28, a1 = - 196.]

| r | = | 1 / 7 | = 1 / 7 < 1, so the series converges and the sum exists.

S = a11– r = - 1961-17
[Use the formula, and replace a1 = - 196, r = 1 / 7.]

S = - 196(7 / 6)
[Simplify.]

S = - 686 / 3

So, the sum of the infinite geometric series is - 686 / 3.


Correct answer : (1)
 6.  
Simplify: 1 - (113)111
a.
12 143
b.
12 11
c.
132 13
d.
132


Solution:

1 - (113)111 = 143(1 - (113))143(111)
[Multiply the numerator and denominator by 143, the LCD of 11 and 13.]

= 143-11 / 13 = 132 / 13.
[Simplify.]


Correct answer : (3)
 7.  
Simplify: 6 + (14)724
a.
43 24
b.
150 7
c.
145 7
d.
25 4


Solution:

6 + (14)724

= 24(6 + (14))24(724)
[Multiply numerator and denominator with 24, the LCD of 4 and 24.]

= 144+6 / 7 = 150 / 7.
[Simplify.]


Correct answer : (2)
 8.  
Find the sum of the infinite geometric series.
1 + 1 11 + 1 121 + ......
a.
1 11
b.
10 11
c.
11 10
d.
11 12


Solution:

r = a2a1 = 1111 = 1 / 11
[The common ratio.]

| r | = | 1 / 11 | = 1 / 11 < 1, so the series converges and the sum exists.

S = a11-r= 11 - (111)
[Use the formula, and replace a1 = 1, r = 1 / 11.]

S = 1(1011) = 11 / 10
[Simplify.]

So, the sum of the infinite geometric series is 11 / 10.


Correct answer : (3)
 9.  
Find the sum of the infinite geometric series.
1 - 1 12 + 1 144 - ......
a.
12 13
b.
12 11
c.
11 12
d.
13 12


Solution:

r = a2a1 = - 1121 = - 1 / 12
[The common ratio.]

| r | = | - 1 / 12 | = 1 / 12 < 1, so the series converges and the sum exists.

S = a11-r = 11 - (- 112)
[Use the formula, and replace a1 = 1, r = - 1 / 12.]

S = 1(1312) = 12 / 13
[Simplify.]

So, the sum of the infinite geometric series is 12 / 13.


Correct answer : (1)
 10.  
Find the sum of the infinite geometric series.
- 30 + 1 - 1 30 + .......
a.
- 900 31
b.
- 871 30
c.
- 1 31
d.
- 60 31


Solution:

r = a2a1 = 1- 30 = - 1 / 30
[Replace a1 = - 30 and a2 = 1.]

| r | = | - 1 / 30 | = 1 / 30 < 1, so the series converges and the sum exists.

S = a11-r = - 301-(- 130)
[Use the formula and replace a1 = - 30, r = - 1 / 30.]

S = - 900 / 31
[Simplify.]

So, the sum of the infinite geometric series is - 900 / 31.


Correct answer : (1)

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