Find the series, then we'll talk

I could find out that the series went something like this:

$1\times \dfrac{1(1+1)}{2} + 2\times \dfrac{2(2+1)}{2} + 3 \times \dfrac{3(3+1)}{2} + \ldots$

So, by inference, we can realize that the $n^{th}$ term (denoted here by $t_n$ ) is:

$t_n = n\times \dfrac{n(n+1)}{2}$

We need to find

$\displaystyle \sum_{n=1}^{50} \dfrac{n^3 + n^2}{2} = S$

$\Rightarrow S = \displaystyle \sum_{n=1}^{50} \dfrac{n^3}{2} + \sum_{n=1}^{50} \dfrac{n^2}{2}$

$\Rightarrow 2S = \left(\dfrac{50\times 51}{2}\right)^2 + \dfrac{50 \times 51 \times 101}{6} = 1625625 + 42925 = 1668550$

$\Rightarrow S = \dfrac{1668550}{2} = \boxed{834275}$

Use this java programme

class brilliant3
{
public static void main(String args[])
{
int x=0;
for(int i=1,f=1;i<=50;i++,f+=i)
{
x+=i*f;
}
System.out.println("sum is"+x);
}
}