Question

Sum of n terms of series $1.3+3.5+5.7+$__________ is?

• A. $1/3n(n+1)(2n+1)+n$
• B. $2/3n(n+1)(2n+1)-n$
• C. $2/3n(n-1)(2n-1)-n$
• D. None of these

Answer 1

Answer: (A)

$1/3n(n+1)(2n+1)+n$

$T_n=(2n-1)(2n+1)={4n}^2-1$

$S_n=\sum T_n=\sum ({4n}^2-1)=4\sum n^2-\sum \left(1\right)$

$S_n=\frac{4n(n+1)(2n+1)}{6}-\frac{n(n+1)}{2}$

$S_n=\frac{n(n+1)}{2}[\frac{8n+4}{3}-\frac{3}{3}]$

$S_n=\frac{n(n+1)(8n+1)}{6}$