Question

The sum to the first n terms of the series $\frac{1}{2}+\frac{3}{4}+\frac{7}{8}+\frac{15}{16}+\dots$ is $9+2^{-10}$. The value of n is

• A. $10$
• B. $9$
• C. $8$
• D. none of these

Answer 1

Answer: (A)

$10$

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

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

$=(1+1+1+.......uptonterms)-(\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+.....nterms)$

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

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

Now,

$n-1+2^{-n}=9+2^{-10}$

$\Rightarrow n=10$