iGCSE (2021 Edition)

# 9.06 Infinite geometric series

Worksheet
Infinite geometric series
1

What condition must be satisfied by an infinite geometric series in order for its sum to exist?

2

For each of the following infinite geometric sequences:

i

Find the common ratio.

ii

Determine whether the sum is divergent or convergent.

a

11, \, 22, \, 44, \, 88, \ldots

b

3, \, -12, \, 48, \, -192, \ldots

c

-40, \, -20, \, -10, \, -5, \ldots

d

256, \, 64, \, 16, \, 4, \ldots

3

For each of the following infinite geometric sequences:

i

Find the common ratio, r.

ii

Find the limiting sum of the series.

a

2, \, \dfrac{1}{2}, \, \dfrac{1}{8}, \, \dfrac{1}{32}, \ldots

b

125, \, 25, \, 5, \, 1, \ldots

c

16, \, - 8, \, 4, \, - 2, \ldots

d

28, \, - 7, \, \dfrac{7}{4}, \, - \dfrac{7}{16}, \ldots

4

Consider the infinite geometric series: 5 + \sqrt{5} + 1 + \ldots

a

Find the common ratio, r, expressing your answer with a rational denominator.

b

5

Consider the infinite geometric series: 6 + 2 + \dfrac{2}{3} + \dfrac{2}{9} + \ldots

a

Find the number of terms, n, that would be required to give a sum of \dfrac{177\,146}{19\,683}.

b

Find the infinite sum of the series.

6

Find the limiting sum of the infinite series \dfrac{1}{5} + \dfrac{3}{5^{2}} + \dfrac{1}{5^{3}} + \dfrac{3}{5^{4}} + \dfrac{1}{5^{5}} + \ldots

7

For a particular geometric sequence, a = 2 and S_{\infty} = 4.

a

Find r, the common ratio.

b

Find the first 3 terms in the sequence.

8

Consider the recurring decimal 0.2222 \ldots in the form \dfrac{2}{10} + \dfrac{2}{100} + \dfrac{2}{1000} + \dfrac{2}{10\,000} + \ldots Rewrite the recurring decimal as a fraction.

9

Find the value of the following:

a

\sum_{i=1}^{\infty} 3 \left(\dfrac{1}{4}\right)^{i - 1}

b

\sum_{i=1}^{\infty} 5 \left( - \dfrac{1}{4} \right)^{i - 1}

c

\sum_{i=1}^{\infty} \dfrac{1}{8} \left( - \dfrac{1}{2} \right)^{i - 1}

d

\sum_{i=1}^{\infty} - \dfrac{1}{2} \left(\dfrac{5}{7}\right)^{i - 1}

e

\sum_{i=1}^{\infty} \left(0.9\right)^{i}

f

\sum_{k=1}^{\infty} 9^{ - k }

Recurring decimals
10

For each of the following recurring decimals:

i

Rewrite the decimal as a geometric series.

ii

Express the decimal as a fraction.

a

0.6666 \ldots

b

0.066\,66 \ldots

c

0.151\,515\,15 \ldots

d

0.021\,212\,1 \ldots

11

Consider the recurring decimal 0.575\,75 \ldots in the form 0.57 + 0.0057 + \ldots Rewrite the recurring decimal as a fraction.

12

Consider the sum 0.25 + 0.0025 + 0.000\,025 + \ldots

a

Express the sum as a decimal.

b

Hence, express the decimal as a fraction.

13

Find the value of 0.994 + 0.000\,994 + 0.000\,000\,994 \ldots

Algebraic series
14

Consider a geometric series in which all terms are positive that has first term a and common ratio r. The sum of the infinite series is 768. The sum of the first 4 terms is 765.

Find the first three terms of the sequence.

15

The limiting sum of the infinite sequence 1, \, 4x, \, 16 x^{2}, \ldots is 5. Find the value of x.

16

Consider the infinite sequence: 1, x - 7, \left(x - 7\right)^{2}, \left(x - 7\right)^{3}, \ldots

Determine the range of values of x if the infinite series has a limiting sum.