Today, I saw a problem on the Mathematics Research and Development Forum:
\sum_{j=0}^{n} (jx^j)=\frac{nx^{n+2}-(n+1)x^{n+1}+x}{(x-1)^2}
This problem is actually just finding the summation formula for x+2x^2+3x^3+\dots+nx^n.
Originally, using mathematical induction would be very simple (mathematical induction is easy for proof but not for derivation), but the problem states that mathematical induction cannot be used. Therefore, the following method is employed.
We rewrite it as:
\begin{aligned} & x(1+2x+3x^2+\dots+nx^{n-1}) \\ \Rightarrow & x[1+x+x^2+\dots+x^{n-1}+x(1+x+x^2+\dots+x^{n-2})+\dots+x^{n-1}] \\ \Rightarrow & x\left[\frac{\frac{x^n-1}{x-1}+x\frac{x^{n-1}-1}{x-1}+x^2\frac{x^{n-2}-1}{x-1}+\dots+x^{n-1}\frac{x-1}{x-1}}{1}\right] \\ \Rightarrow & x\left[\frac{nx^n-(1+x+x^2+\dots+x^{n-1})}{x-1}\right] \\ \Rightarrow & x\left[\frac{nx^n-\frac{x^n-1}{x-1}}{x-1}\right] \end{aligned}
\Rightarrow x\left[\frac{nx^{n+1}-(n+1)x^n+1}{(x-1)^2}\right]
The rest is omitted. It should be self-explanatory.
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