So...Many...Sums

The question is equivalent to asking for the sum of the first n triangular numbers. You can prove by induction that the sum of the first n triangular numbers is:

$\frac{n(n+1)(n+2)}{6}$

In this case, we have n = 100 and thus the solution is:

$\frac{100(100+1)(100+2)}{6}$ = 171700

directly put n=100 in the equation 1/6(n)(n+1)(n+2). This is the short method for such questions!

The sum of the first x integers is x(x+1)/2. This is also equivalent to $\binom{x+1}{2}$ . So our sum of sums is in fact equivalent to $\binom{2}{2}$ + $\binom{3}{2}$ + ... + $\binom{101}{2}$ . The Hockey Stick Identity states that this sum is equal to $\binom{102}{3}$ = 171700. So the last 3 digits are 700.