So I know that you can take the derivative of this and multiply by x and do integrals or something like that. However, I am just wondering if there is a way to come to the summation of the series without doing derivatives or integrals. Is there a simple way or would that way just be guess and check?
2026-03-25 11:15:06.1774437306
How to calculate the summation of $n \cdot 2^n$?
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$$4\cdot2^4+3\cdot2^3+2\cdot2^2+1\cdot2^1 \\=(4\cdot2^5+3\cdot2^4+2\cdot2^3+1\cdot2^2)-(4\cdot2^4+3\cdot2^3+2\cdot2^2+1\cdot2^1) \\=4\cdot2^5-2^4-2^3-2^2-2^1 \\=4\cdot2^{4+1}-2^{4+1}+2.$$
The generalization is immediate.