Question

Find the sum of n terms of the series $\mathrm{1.3.5}+\mathrm{3.5.7}+\mathrm{5.7.9}+....$. The $n^{th}$ term is $(2n-1)(2n+1)(2n+3);$ hence by the rule $S_n=\frac{(2n-1)(2n+1)(2n+3)(2n+5)}{4.2}+c$.

Answer 1

o determine C, put $n=1;$ then the series reduces to its first term, and we have
$15=\frac{\mathrm{1.3.5.7}}{8}+C;$ hence $C=\frac{15}{8};$
$\therefore S_n=\frac{(2n-1)(2n+1)(2n+3)(2n+5)}{8}+\frac{15}{8}$
$=n(2n^3+8n^2+7n-2),$ after reduction.