Question

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

• A. $2^n-n-1$
• B. $1-2^{-n}$
• C. $2^{-n}+n-1$
• D. None of these

Answer 1

Answer: (C)

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

We have,

$S_n=\frac{1}{2}+\frac{3}{4}+\frac{7}{8}+\frac{15}{16}+...$

$=(1-\frac{1}{2})+(1-\frac{1}{4})+(1-\frac{1}{8})+...+(1-\frac{1}{2^n})$

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

$=n-\frac{\frac{1}{2}[1-{\left(\frac{1}{2}\right)}^n]}{1-\frac{1}{2}}=n-1+2^{-n}$