Question

Find the general term and the sum of n terms of the series $4,13,35,94,262,755,....$

Answer 1

$\Rightarrow 4133594262\mathrm{755....}$

$92259168\mathrm{498....}$

$1337109\mathrm{325....}$

$2472\mathrm{216....}$

$\therefore u_n=a.3^{n-1}+b{n}^2+cn+d$

Putting $n=1,2,3,4$, we find;-

$a=3,b=\frac{1}{2},c\frac{3}{2},d=-1$

$u_n=3.3^{n-1}+\frac{1}{2}{n}^2+\frac{3}{2}n-1=3^n-1+\frac{1}{2}n(n+3)$

$S_n=\sum u_n$

$=\frac{3}{2}(3^n-1)-n+\frac{n(n+1)(2n+1)}{12}+\frac{3}{2}\frac{n(n+1)}{2}$

$=\frac{1}{2}(3^{n+1}-3)+\frac{n(n+1)(n+5)}{6}-n$