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Sequence and Series

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Question : 40 of 86

Marks: +1, -0

The sum of the first two terms of an infinite geometric series is 15 and each term is equal to the sum of all the terms following it; then the sum of the series is

20
15
25
None

Solution:

Concept:

Use the property of infinite geometric series: each term equals sum of all subsequent terms gives

r = 1/2

.

Explanation:

Let the series be

a, ar, ar^2, \ldots

with

|r|<1

.

Given:

a + ar = 15

.

Also, each term equals sum of all terms after it: for the first term,

a = ar + ar^2 + \cdots = \frac{ar}{1-r}

.

Cancel

a

(

a \neq 0

):

1 = \frac{r}{1-r} \Rightarrow 1-r = r \Rightarrow r = \frac{1}{2}

.

From

a(1+r)=15

:

a\left(1+\frac{1}{2}\right)=15 \Rightarrow a \cdot \frac{3}{2}=15 \Rightarrow a=10

.

Sum of infinite series:

S = \frac{a}{1-r} = \frac{10}{1-\frac{1}{2}} = \frac{10}{\frac{1}{2}} = 20

.

Answer:

20

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