Question

If 1 + 2 + 3 + ..... + n = k , then $1^3+2^3+3^3+.........+n^3$ is equal to

• A. $k^2$
• B. $k^3$
• C. $\frac{k(k+1)}{2}$
• D. $(k+1{)}^3$

Answer 1

The correct option is A $k^2$

Since, we know that
$1+2+3+...+n=\frac{n(n+1)}{2}$ ...(i)
And $1^3+2^3+3^3+...+n^3=[\frac{n(n+1)}{2}{]}^2$ ...(ii)
Since, $1+2+3+...+n=\frac{n(n+1)}{2}=k$ ...(iii)
From (i),(ii) and (iii), we get
$1^3+2^3+3^3+...+n^3=k^2$
Option A is correct.