The sum of the first n terms is 12 + 34 + 78 + 1516&#x

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Question

The sum of the first n terms is $\frac{1}{2}$ + $\frac{3}{4}$ + $\frac{7}{8}$ + $\frac{15}{16}$ + .......... is

Options:

A .

2^{n}

- n - 1

B . 1 -

2^{- n}

C . n +

2^{- n}

- 1

D .

2^{n}

- 1

Answer: Option C
:
C

The sum of the first n terms is

$S_{n}$ = (1 - $\frac{1}{2}$ ) + (1 - $\frac{1}{2^{2}}$ ) + (1 - $\frac{1}{2^{3}}$ ) + (1 - $\frac{1}{2^{4}}$ + ........... + (1 - $\frac{1}{2^{n}}$ )

= n - { $\frac{1}{2}$ + $\frac{1}{2^{2}}$ +............+ $\frac{1}{2^{n}}$ }

= n - $\frac{1}{2}$ ( $\frac{1 - \frac{1}{2^{n}}}{1 - \frac{1}{2}}$ ) = n - (1 - $\frac{1}{2^{n}}$ ) = n - 1 + $2^{- n}$ .

Trick: Check for n = 1, 2 i.e., $S_{1}$ = $\frac{1}{2}$ , $S_{2}$ = $\frac{5}{4}$ and

(c) ⇒ $S_{1}$ = $\frac{1}{2}$ and $S_{2}$ = 2 + $2^{- 2}$ - 1 = $\frac{5}{4}$ .

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