求助三个求和式的化简

lemondian

请大家帮忙化简一下，过程尽量详细一点，谢谢了
\[
\begin{aligned}
& \sum_{k=0}^{r-1} C_{2 r}^k\left[(-1)^{r-k+1}-1\right] \\
& \sum_{k=0}^{r-1}(-1)^{r+k+1} C_{2 r}^k \\
& \sum_{k=0}^{r-1}(-1)^{r+k} C_{2 r}^k\left[(-1)^{r-k+1}-1\right]
\end{aligned}
\]

ZCos666

第三个就是第一个，第二个也包含在第一个中

\[ \begin{aligned}
\sum_{k=0}^{r-1}(-1)^{r+k+1}\binom{2r}{k}&=(-1)^{r+1}\cdot\dfrac{(-1)^{r+1}}{2}\binom{2r}{r}\\
&=\dfrac{1}{2}\binom{2r}{r}
\end{aligned} \]

\[ \begin{aligned}
\sum_{k=0}^{r-1}\left((-1)^{r-k+1}-1\right)\binom{2r}{k}&=\dfrac{1}{2}\binom{2r}{r}-\sum_{k=0}^{r-1}\binom{2r}{k}\\
&=\dfrac{1}{2}\binom{2r}{r}-\dfrac{1}{2}\left(4^r-\binom{2r}{r}\right)\\
&=\binom{2r}{r}-2^{2r-1}
\end{aligned} \]