Hi

I am trying to prove that

$\displaystyle \sum_{k=0}^n (-1)^k \binom{n}{k}=0 \,\, \forall n \in N$

This has to be proved using induction. For n=1, this is certainly true. Now assume

that its true for some $\displaystyle m\in N$. Then we have

$\displaystyle \sum_{k=0}^m (-1)^k \binom{m}{k}=0 $

Now consider P(m+1).

$\displaystyle \sum_{k=0}^{m+1} (-1)^k \binom{m+1}{k}$

$\displaystyle =\sum_{k=0}^{m} (-1)^k \binom{m+1}{k}+(-1)^{m+1}\binom{m+1}{m+1}$

$\displaystyle =(-1)^{m+1}+\sum_{k=0}^{m} (-1)^k \binom{m+1}{k}$

Now I am trying to see where to go from here.

Can anybody help me here ?

newton