Question

Find the general term and sum of $n$ terms of the series $9+16+29+54+103+...$

Answer 1

Let $S=9+16+29+54+103+...+T_n$ (i)
$S=9+16+29+54+103+...+T_{n-1}+T_n$ (ii)
(i)-(ii) $\Rightarrow$ $T_n=9+7+13+25+49+...+(T_n-T_{n-1})$ (iii)
$T_n=9+7+13+25+49+...+(T_{n-1}-T_{n-2})+(T_n-T_{n-1})$ (iv)
(iii)-(iv) $\Rightarrow$ $T_n-T_{n-1}=9+(-2)+6+12+24+...(n-2)terms=7+6[2^{n-2}-1]=6{\left(2\right)}^{n-2}+1$
$\therefore$ General terms is $T_n=6{\left(2\right)}^{n-1}+n+2$
Also sun $S=\sum T_n=6\sum 2^{n-1}+\sum n+\sum 2=6.\frac{2^n-1}{2-1}+\frac{n(n+1)}{2}+2n=6(2^n-1)+\frac{n(n+5)}{2}$