Q65 / 160 · WBJEE 2010 · Mathematics
Sum of $n$ terms of the following series $1^3+3^3+5^3+7^3+\ldots \ldots \ldots$ is
- Correct answerA$n^2\left(2 n^2-1\right)$
- B$\mathrm{n}^3(\mathrm{n}-1)$
- C$n^3+8 n+4$
- D$2 n^4+3 n^2$
Solution
Hints : $\sum(2 n-1)^3$
$$
\begin{aligned}
& \sum\left\{\left(8 n^3-3.4 n^2+3.2 n-1\right)\right\} \\
= & 2 n^2(n+1)^2-2 n(n+1)(2 n+1)+3 n(n+1)-n \\
= & 2 n^4+4 n^3+2 n^2-2 n\left[2 n^2+3 n+1\right]+3 n^2+3 n-n \\
= & 2 n^4+4 n^3+2 n^2-4 n^3-6 n^2-2 n+3 n^2+3 n-n \\
= & 2 n^4-n^2 \\
= & n^2\left(2 n^2-1\right)
\end{aligned}
$$
Hence, the correct answer is option: 'A'.