You remember this one, right? the sum r + r^2 + r^3 + ... + r^n
Surprisingly, it's almost no more difficult.
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?