Question

The sum of $10$ terms of the series $\mathrm{1.3.5}+\mathrm{3.5.7}+\mathrm{5.7.9}+\dots \dots$ is

• A. $28720$
• B. $28780$
• C. $28700$
• D. $28680$

Answer 1

The correct option is D $28680$

The $n^{th}$ term is $(2n-1)(2n+1)(2n+3)$
$T_n=(2n-1)(2n+1)(2n+3)$
$T_n=\frac{1}{8}(2n-1)(2n+1)(2n+3)\left\{\right(2n+5)-(2n-3\left)\right\}$
$=\frac{1}{8}(V_n-V_{n-1})$ $\text{Where}{V}_n=(2n-1)(2n+1)(2n+3)(2n+5)$
$S_n=\sum T_n=\frac{1}{8}[V_n-V_0]$
$\Rightarrow S_n=\frac{(2n-1)(2n+1)(2n+3)(2n+5)}{8}+\frac{15}{8}=n(2n^3+8n^2+7n-2)$
$\therefore S_{10}=28680$