Question

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

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

Answer 1

Answer: (C)

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

Given series is $\frac{1}{2}+\frac{3}{4}+\frac{7}{8}+\frac{15}{16}+.....$
$=(1-\frac{1}{2})+(1-\frac{1}{4})+(1-\frac{1}{8})+(1-\frac{1}{16})+.....$
$=n-\frac{\frac{1}{2}\{1-\frac{1}{2^n}\}}{1-\frac{1}{2}}$

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