Expressing $1-\sum_{k=1}^{r}(-1)^{k-1}\binom{r}{k}$ as one sum?

Question

Hi how can i make this expression into one sum(ie put the 1 inside)

$1-\sum_{k=1}^{r}(-1)^{k-1}\binom{r}{k}$

I know that $\sum_{k=0}^{r}(-1)^{k}\binom{r}{k}$

comes out but i cant follow the steps ?

Answer 1

It's quite simple really:

Expression = $1 - \sum_{k=1}^{r}(-1)^{k-1}\binom{r}{k} = 1 + \sum_{k=1}^{r}(-1)^{k}\binom{r}{k}$ by putting the '-' inside the summation.

Then we also note that if we extend our summation to $k=0$, $(-1)^{0}\binom{r}{0} = 1$. Therefore we can get rid of the $1$ in our expression and extend the summation to include $k=0$.

Expression = $\sum_{k=0}^{r}(-1)^{k}\binom{r}{k}$