Question

Find the sum of 1st n terms of the series 5+7+13+85+.

Answer 1

Consider the given function:

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

$DiffrenceoftermsisinG.P$

$=2,6,\mathrm{18......}$

$Sn=5+3+13+31+85+......+T_n\left(1\right)$

$Sn=5+7+13+31+......T_n\left(2\right)$

$sothan\left(1\right)-\left(2\right)$

$Tn=5+2+6+18+......+(T_n-T_{n-1})$

$Tn=5+2\left(\frac{3^{n-1}-1}{3-1}\right)$

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

$Tn=4+3^{n-1}$

$S_n=\sum _{k=1}^n{T}_k=\sum _{k=1}^{n}4+3^{k-1}$

$S_n=\sum _{k=1}^{n}4+\sum _{k=1}^n{3}^{k-1}$

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

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

$S_n=4n+\frac{3^n-1}{2}$

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

Hence this is the answer.