[ Sums of numerical sequences and series of differences ] Pascal's Triangle and the Binomial Theorem Complex numbers help solve tasks_

Using the expansion $(1+i)^{n}$ by the Binomial Theorem, find:

a) $C_{100}^{0}-C_{100}^{2}+C_{100}^{4}-\ldots+C_{100}^{100}$

b) $C_{99}^{1}-C_{99}^{3}+C_{99}^{5}-\ldots-C_{99}^{99}$.
a) The given sum is equal to $\operatorname{Re}(1+i)^{100}$, and $(1+i)^{100}=(2 i)^{50}=-2^{50}$.

b) The sum is equal to $\operatorname{Im}(1+i)^{99}$, and $(1+i)^{99}=(1+i)(2 i)^{49}=(1+i) \cdot 2^{49} i^{49}=(1-i) \cdot 2^{49}$.

## Answer

a) $-2^{50}$;
b) $2^{49}$.

## Problem