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JAMB Mathematics 2010 Paper

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Question : 35

Total: 49

Find the sum to infinity of the following series. 0.5 + 0.05 + 0.005 + 0.0005 + .....

$\frac{5}{8}$
$\frac{5}{7}$
$\frac{5}{11}$
$\frac{5}{9}$

Solution:

Concept:

This is an infinite geometric series. Use the formula

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

.

Explanation:

The first term is

a = 0.5 = \frac{1}{2}

.

The common ratio is

r = \frac{0.05}{0.5} = 0.1 = \frac{1}{10}

.

Since

|r| < 1

, the series converges.

Substitute into the formula:

S_{\infty} = \frac{0.5}{1 - 0.1}

.

This gives

S_{\infty} = \frac{0.5}{0.9}

.

Simplify:

S_{\infty} = \frac{\frac{1}{2}}{\frac{9}{10}} = \frac{1}{2} \times \frac{10}{9} = \frac{5}{9}

.

Answer:

The sum to infinity is

\frac{5}{9}

, which is Option D.

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