Question

The sum of $40$ terms of the series $1+2+3+4+5+8+7+16+9+......$ is,

• A. $398+2^{20}$
• B. $398+2^{21}$
• C. $398+2^{19}$
• D. $Noneofthese$

Answer 1

Answer: (B)

$398+2^{21}$

The given series $X=1+2+3+4+5+8+7+16+9+....$

This is sum of $2$ series $X_1$ and $X_2$

$X_1+X_2=X$

$X_1=1+3+5+7+9+.......$

$X_2=2+4+8+16+.......$

$X_1$ is an AP with first term $1$ and common difference $2$

$X_2$ is a GP with first term $2$ and common ratio $2$

Now for sum up to $40$ term for $X$, we need to find sum of first $20$ terms of the GP and first $20$ terms of AP.

Sum of AP $=S_{AP}$ =$\frac{20}{2}(2+(20-1)2)=10(2+38)=400$

Sum of GP $=S_{GP}=\frac{2(1-2^{20})}{1-2}=-2+2^{21}$

So, Sum $X$ $=400-2+2^{21}$ $=$ $398+2^{21}$