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Sequence and Series

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Question : 69 of 86

Marks: +1, -0

A radioactive sample decays and the remaining sample at infinite time is given by

b = 1 - \left(\;\frac{1}{2} + \;\frac{1}{4} + \dots \to \infty\right),

b

is

0
1
$\;\frac{1}{\sqrt{2}}$
$\;\frac{1}{2}$

Solution:

Concept:

The sum of an infinite geometric series is used to evaluate the given expression.

Explanation:

The series inside the parentheses is

\frac{1}{2} + \frac{1}{4} + \dots

to infinity.

This is a geometric progression with first term

a = \frac{1}{2}

and common ratio

r = \frac{1}{2}

.

Sum to infinity

S_\infty = \frac{a}{1-r} = \frac{1/2}{1-1/2} = \frac{1/2}{1/2} = 1

.

Thus

b = 1 - S_\infty = 1 - 1 = 0

.

Answer:

0

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