Question

The A.M. of first n terms of the series 1.3.5, 3.5.7, 5.7.9,....., is

• A. $3n^3+6n^2+7n-1$
• B. $n^3+8n^2+7n-1$
• C. $2n^3+8n^{2}7n-2$
• D. $2n^3+8n^2+7n-2$

Answer 1

The correct option is D $2n^3+8n^2+7n-2$

$\because T_n=(2n-1)(2n+1)(2n+3)$

Req. A.M.

$=\frac{S_n}{n}=\frac{\sum T_n}{n}=\frac{1}{n}[\sum (8n^3+12n^2-2n-3\left)\right]$

$=\frac{1}{n}[8\sum n^3+12\sum n^2-2\sum n-\sum 3]$

$=2n^3+8n^2+7n-2$