Question

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

• A. $2^n$ - n - 1
• B. 1 - $2^{-n}$
• C. n + $2^{-n}$ - 1
• D. $2^n$ - 1

Answer 1

The correct option is C

n + $2^{-n}$ - 1

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 = $\frac{5}{4}$.