Question

Let $S=1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+...$ find the sum of first $20$ terms of the series

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

Answer 1

The correct option is C $\frac{2^{20}-1}{2^{19}}$

$S=1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}.......$ first $20$ terms

$n=20$ and series is in $GP$ with common difference $=\frac{\frac{1}{2}}{1}=\frac{\frac{1}{4}}{\frac{1}{2}}=\frac{1}{2}$

$a=1$ and $r=\frac{1}{2}$

Sum$=\frac{a(1-r^n)}{1-r}$ when$r<1$

$=\frac{1(1-{\left(\frac{1}{2}\right)}^{20})}{1-\frac{1}{2}}=\frac{(1-\frac{1}{2^{20}})}{\frac{1}{2}}=2(1-\frac{1}{2^{20}})=\frac{2(2^{20}-1)}{2^{20}}=\left(\frac{2^{20}-1}{2^{19}}\right)$