# Thread: Prove this .

1. ## Prove this .

Prove that $\displaystyle \sum^{n}_{r=1}(r+1)2^{r-1}=n2^n$ .

2. Originally Posted by thereddevils
Prove that $\displaystyle \sum^{n}_{r=1}(r+1)2^{r-1}=n2^n$ .
Telescoping series:

$\displaystyle (r+1)2^{r-1}=r2^r-(r-1)2^{r-1}$

CB

3. Thanks Captain , but how did you get that expression ? Is it by expansion ?

4. Originally Posted by thereddevils
Thanks Captain , but how did you get that expression ? Is it by expansion ?
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

5. I know what a telescoping series is but i just don understand how you get this :

$\displaystyle (r+1)2^{r-1}=r2^r-(r-1)2^{r-1}$

6. Originally Posted by thereddevils
I know what a telescoping series is but i just don understand how you get this :

$\displaystyle (r+1)2^{r-1}=r2^r-(r-1)2^{r-1}$
$\displaystyle r 2^{r-1} + 2^{r-1} = r 2^{r-1} (2 - 1) + 2^{r-1} = r 2^r - r 2^{r - 1} + 2^{r-1} = r 2^r - 2^{r-1} (r - 1)$.