Prove that $\displaystyle \sum^{n}_{r=1}(r+1)2^{r-1}=n2^n$ .
First I deduce what method is appropriate for high school level, which made me suspect a telescoping series, then noting that $\displaystyle 0\times 2^0=0$ is a give away for writting the general term as $\displaystyle r2^r-(r-1)2^{r-1}$, since then the only part of the last term of the series that does not cancel is $\displaystyle n2^2$, and from the first term is $\displaystyle 0$.
CB