Question

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

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

Answer 1

Answer: (A)

Given:
$1+\frac{3}{2}+\frac{7}{4}+ \frac{15}{8}+\frac{31}{6}+\cdots$
$=(2-1)+\left(2-\frac{1}{2}\right)+\left(2-\frac{1}{4}\right)+\left(2-\frac{1}{8}\right)+\left(2-\frac{1}{16}\right)+\dots$ up to 20 terms
$=40-(1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\dots$ up to 20 terms )
$=40-\left(\frac{1-\left(\frac{1}{2}\right)^{20}}{1-\frac{1}{2}}\right)$
$=40-2+\frac{1}{2^{19}}=38+\frac{1}{2^{19}}$