Question

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

• A.
• B.
• C.
• D.

Answer 1

The correct option is D

$S=\frac{1}{2}+\frac{3}{4}+\frac{7}{8}+\frac{15}{16}+....$ to 'n' terms
$S=(1-\frac{1}{2})+(1-\frac{1}{4})+(1-\frac{1}{8})+.......$ to 'n' terms
$=(1+1+1+....)-(\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+.....)=n-\frac{1}{2}\frac{1-{\left(\frac{1}{2}\right)}^n}{2-\frac{1}{2}}=n-(1-\frac{1}{2^n})=2^{-n}+n-1$