Triangle Method for Weighted Sums

Idea · Intermediate

Algebra: Sequences & Series

Start with:

1\cdot 2^1+2\cdot 2^2+3\cdot 2^3+\cdots+(n-1)2^{n-1}+n2^n

Write the following equations in a triangular arrangement:

Then add them:

n2^{n+1}-\left(2^1+2^2+2^3+\cdots+2^{n-1}+2^n\right)

Since

2^1+2^2+\cdots+2^{n-1}+2^n=2^{n+1}-2^1

we get

n2^{n+1}-\left(2^{n+1}-2^1\right)

\boxed{(n-1)2^{n+1}+2^1}