Calculate the summations
Hello, WORLD!
This is a tough problem for the algebra forum. I'm going to use a little calculus. I hope you will still find my remarks useful.
Start with the Binomial Theorem:
Integrate both sides from 0 to x (this is the calculus part). The result is
Now multiply these two equations together. I'm going to replace i with j in the first equation, so it will be easier to keep everything straight.
(*)....
Now consider the coefficient of in (*). The left-hand side of (*) is equal to
so the coefficient of is
On the right-hand side of (*), the coefficient of occurs when j = n-i, so the coefficient is
since .
The coefficient of must be the same on both sides of the equation, so we have
Now set n = 100, and you're done.