Question

Find the sum od series upto n terms 5+7+13+31+85+......

Answer 1

Please see the attachment

Answer 2

Answer:

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

Step-by-step explanation:

Let the series sum be Sₙ and nth term is tₙ

$S_n=5+7+13+31+85+............................+n-terms$

$S_n=0+5+7+13+31+85+............................+n-terms$

Subtract both equation

$0=5+(2+6+18+54+............................+(n-1)-terms)-t_n$

2+6+18+54+............................+(n-1)-terms   This is GP series

Using sum of series of geometric series

$t_n=5+\dfrac{2(3^{n-1}-1)}{3-1}$

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

So, the general term of the above series is tₙ

Now, we find the sum of ∑tₙ

$S_n=\sum t_n$

$S_n=\sum_1^n(4+3^{n-1})$

$S_n=\sum_1^n4+\sum_1^n3^{n-1}$

$S_n=4n+\dfrac{1\cdot(3^{n}-1)}{3-1}$

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

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