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[tex] S_n = \sum_{i=0}^{n-1} i2^i [/tex]

I tried the following. Let r = 2, and figure out the terms in

[tex] S_n - rS_n [/tex]

Unlike with a regular geometric series, this does not make all but two of the terms disappear. But it does make all but one of the terms turn into a simple power of 2 (once you collect like powers of 2). In other words, it turns into something plus a regular geometric series. For my final answer, solving for S_n, I got:

[tex] S_n = (n-2)2^n + 2 [/tex]

but I have reason to believe this is incorrect. Can anybody help me out?