Question

What is the sum of $1^3 + 2^3 + 3^3 +....+n^3$ ?

• A. $\left[\frac{n\left(n+1\right)}{3}\right]^{2}$
• B. $\left[\frac{n\left(n+1\right)}{2}\right]^{2}$
• C. $\left[\frac{n\left(n+2\right)}{3}\right]^{2}$
• D. $\left[\frac{n\left(n+1\right)}{2}\right]^{3}$

Answer 1

Answer: (B)

$P\left(k\right) = 1^{3}+2^{3}+3^{3} +...+k^{3} = \left\{\frac{k\left(k+1\right)}{2}\right\}^{2}$
Now, $ \left(1^{3} + 2^{3} +3^{3} +...+k^{3}\right)+\left(k+1\right)^{3} $
$= \left\{ \frac{k\left(k+1\right)}{2}\right\}^{2} + \left(k+1\right)^{3}$
$ = \frac{1}{4} k^{2}\left(k+1\right)^{2} + \left(k+1\right)^{3}$
$ = \frac{1}{4} \left[k^{2} \left(k+1\right)^{2} + 4\left(k+1\right)^{3}\right]$
$ = \frac{1}{4}\left(k+1\right)^{2} \left[k^{2}+4\left(k+1\right)\right] $
$= \frac{1}{4} \left(k+1\right)^{2} + \left(k+2\right)^{2} $