Question

Sum of the series to $n$ terms $5+7+13+31+85+\cdots$ is

• A. $3^n+8n+1$
• B. $\frac{1}{2}[3^n+8n-1]$
• C. $\frac{1}{2}(3^n+8n+1)$
• D. None of the above

Answer 1

The correct option is B $\frac{1}{2}[3^n+8n-1]$

The sequence of difference between successive terms is $2,6,18,54,\dots$
Clearly, it is a GP. Let $T_n$ be the $n^{th}$ term of the given series and $S_n$ be the sum of its $n$ terms.
Then, $S_n=5+7+13+85+\cdots +T_{n-1}+T_n$ ...(i)
$S_n=5+7+13+31+\cdots +T_{n-1}+T_n$ ...(ii)
Subtracting (ii) from (i), we get
$0=5+[2+6+18+54+\cdots +(T_n-T_{n-1})]-T_n$
$\Rightarrow 0=5+2\cdot \frac{3^{n-1}-1}{3-1}-T_n$
$\Rightarrow T_n=5+(3^{n-1}-1)=4+3^{n-1}$
Therefore, $S_n=\sum _{k=1}^n{T}_k=\sum _{k=1}^n(4+3^{k-1})$
$=\sum _{k=1}^{n}4+\sum _{k=1}^n{3}^{k-1}$
$=4n+(1+3+3^2+\cdots +3^{n-1})$
$=4n+1\times \left(\frac{3^n-1}{3-1}\right)$
$=4n+\left(\frac{3^n-1}{2}\right)$
$=\frac{1}{2}(3^n+8n-1)$