What have you tried? It looks trickier than it is.

You remember this one, right? the sum r + r^2 + r^3 + ... + r^n

Surprisingly, it's almost no more difficult.

Your series:

r + 2r^2 + 3r^3 + ... + nr^n

Give it a Sum

1) r + 2r^2 + 3r^3 + ... + nr^n = S

Multiply by r

2) r^2 + 2r^3 + 3r^4 + ... + nr^(n+1) = Sr

Subtract 2) from 1)

r + r^2 + 3r^3 + ... + nr^n - nr^(n+1) = S - Sr

It doesn't look like we're getting anywhere, does it? Before we give up, let's just group it a little.

[r + r^2 + 3r^3 + ... + nr^n] - nr^(n+1) = S - Sr

What does that part in the brackets look like? Familiar?