Divisibility by 11 I'm trying to prove that a number $a_n=a_0+a_1 \cdot 10^1+...+c_n\cdot 10^n$ is divisible by 11 iff $a_0-a_1+a_2-...+(a_n)\cdot(-1)^{n+1}$ is divisible by 11.
I began my proof and I found out that I first need to prove that $10^k+(-1)^{k+1}$ is divisible by 11 for all $k\in \mathbb{N} $.

I tried to prove this with induction:
for $k=1$ we get $t_1=10^1+(-1)^{1+1}=11$
now I assume $t_k=10^k+(-1)^{k+1}$ is divisible by 11.
need to prove that: $\\$ $t_{k+1}=10^{k+1}+(-1)^{k+2}=10^{k+1}+(-1)^{k}$
is divisible by 11.

I looked at:

$t_{k+1}-t_{k}=10^{k+1}+(-1)^{k}-10^k-(-1)^{k+1}=9\cdot 10^k +2\cdot (-1)^k$

and now I'm stuck :(
if you have any idea how can I complete my proof I will be very grateful!

Thank You All!

Note first that $$y^k-x^k = (y-x)\sum_{m=0}^{k} y^m x^{k-m}$$
(you can prove this for yourself as the right hand side is a telescoping sum). Then $$10^k + (-1)^{k+1} = 11 \sum_{m=0}^{k} (-1)^{k-m} 10^m$$
which is divisible by 11.
