The value which is not possible for $\sum_{k=1}^n k^3$ is 
a)25
b)36
c)225
d)441

The value becomes $$\left[\frac{(n)(n+1)}{2}\right]^2$$
Since all options are perfect squares, I don’t really know what to do with This.
The discrimaint ie. $b^2-4ac>0$ for all options if we solve the quadratic equation for each option.
The right answer is $25$. How do we get that?

Let
$$\left[\frac{(n)(n+1)}{2}\right]^2=x$$
It is quadratic in $n$.
$$n^2+n-2\sqrt{x}=0$$
The discriminant $$1+4*1*2\sqrt{x}=1+8\sqrt{x}$$ should be a perfect square because $n$ must be whole number.
So, number need not only be a square number. It should satisfy the above condition as well.
$25$ doesn't satisfy it and hence the answer.
