1. ## Summation problem

I am supposed to find a formula for the sum r+2r^2 + 3r^3 + ... + nr^n. I know that it's the summation of k=1 to n of kr^k, but I don't know the equivalent formula to that. Can anyone help?

2. Originally Posted by dnlstffrd
I am supposed to find a formula for the sum r+2r^2 + 3r^3 + ... + nr^n. I know that it's the summation of k=1 to n of kr^k, but I don't know the equivalent formula to that. Can anyone help?
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.

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?

3. Well obviously it looks like the original sum, but how does that help me to find a generalized formula for the sum?

4. You have to reach out and grab these. Don't just answer the question. Think about what the answer might mean and if there are any important implications?

What is the formula for the sum in the brackets? Is it a sufficiently simple expression that you could substitute in the brackets and still manage to solve for S?

Go for it!

5. We have the identity:
$\displaystyle 1+x+x^2+x^3+\ldots +x^n=\frac{x^{n+1}-1}{x-1}$
Take derivative of both sides:
$\displaystyle 1+2x+3x^2+\ldots+nx^{n-1}=\left(\frac{x^{n+1}-1}{x-1}\right)'=\frac{nx^{n+1}-(n+1)x^n+1}{(x-1)^2}$
Now multiply both sides by $x$ and you're done.

6. ...and thus we see the value of posting in some section where we can get clue how you are supposed to proceed. For this example, if you post in the calculus section, we can talk a lot less about algebra.