Question

Sum of the first $20$ terms of the series
$1+\frac{3}{2}+\frac{7}{4}+\frac{15}{8}+\frac{31}{16}+\cdots$

• A. $38+\frac{1}{2^{19}}$
• B. $40+\frac{1}{2^{19}}$
• C. $38+\frac{1}{2^{18}}$
• D. $40+\frac{1}{2^{18}}$

Answer 1

The correct option is A $38+\frac{1}{2^{19}}$

Let
$S_{20}=1+\frac{3}{2}+\frac{7}{4}+\frac{15}{8}+\frac{31}{16}+\cdots \text{upto 20 terms}\Rightarrow S_{20}=(2-1)+(2-\frac{1}{2})+(2-\frac{1}{4})+(2-\frac{1}{8})+\cdots \text{upto 20 terms}\Rightarrow S_{20}=40-[1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\cdots \text{upto 20 terms}]=40-\left[\frac{(1/2{)}^{20}-1}{\left(1/2\right)-1}\right]=40-2+\frac{1}{2^{19}}=38+\frac{1}{2^{19}}$