Question

Find the $n^{th}$ term and the sum of n terms of the series $4,14,30,52,80,114,....$

Answer 1

To find the $n^{th}$ term we have to look the succesive difference of the series

$4,14,30,52,80,\mathrm{114.....}10,16,22,28,\mathrm{34....}6,6,6,\mathrm{6.....}$

Let $u_n=A+Bn+C{n}^2$

For $n=1$ $u_1=4$

$\Rightarrow A+B+C=\mathrm{4.......}\left(i\right)$

For $n=2$ $u_2=14$

$\Rightarrow A+2B+4C=\mathrm{14.......}\left(ii\right)$

For $n=3$ $u_3=30$

$\Rightarrow A+3B+9C=\mathrm{30......}\left(iii\right)$

Solving $\left(i\right),\left(ii\right)$ and $\left(iii\right)$

$\Rightarrow A=0,B=1,C=3$

So the $n^{th}$ term is

$u_n=0+\left(1\right)n+\left(3\right)n^2{u}_n=n+3n^2$