Question

The sum of the first $20$ terms of the series
$1+\frac{3}{2}+\frac{7}{4}+\frac{15}{8}+\frac{31}{16}+....,$ is

• A. $39+\frac{1}{2^{19}}$
• B. $38+\frac{1}{2^{20}}$
• C. $38+\frac{1}{2^{19}}$
• D. $39+\frac{1}{2^{20}}$

Answer 1

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

$1+\frac{3}{2}+\frac{7}{4}+\frac{15}{8}+\frac{31}{16}+....\text{upto 20 terms}=(2-1)+(2-\frac{1}{2})+(2-\frac{1}{4})+(2-\frac{1}{8})+(2-\frac{1}{16})+....\text{upto 20 terms}=40-(1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+....\text{upto 20 terms})=40-\left(\frac{1-{\left(\frac{1}{2}\right)}^{20}}{1-\frac{1}{2}}\right)=40-2+\frac{2}{2^{20}}=38+\frac{1}{2^{19}}$