Question

Sum of the following series to n terms :

$5+7+13+31+85+...$

Answer 1

The sequence of differences between successive terms is $2,6,18,54,...$

Clearly, it is a GP. Let $T_n$ be the nth term of the given series and $S_n$ be the sum of its n terms. Then,

$S_n=5+7+13+31+85+...+T_{n-1}+T_n$ .....(1)

Also, $S_n=0+5+7+13+31+...+T_{n-1}+T_n$ ...(2)

Subtracting 2 from 1 , we get,

$0=5+[2+6+18+54+....+(T_n-T_{n-1}\left)\right]-T_n$

$0=5+\frac{2(3^{n-1}-1)}{(3-1)}-T_n$

$T_n=5+(3^{n-1}-1)=4+3^{n-1}$

Therefore,

$S_n=4n+(1+3+3^2+....+3^{n-1})$

$S_n=4n+1\times \left(\frac{3^n-1}{3-1}\right)$

$S_n=4n+\left(\frac{3^n-1}{2}\right)=\frac{1}{2}(3^n+8n-1)$