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!

Hint
$$
10^{k+1}+(-1)^{k+2}=10^{k+1}+(-1)^{k+1}\cdot 10-(-1)^{k+1}\cdot 11=10 \cdot \left(10^{k}+(-1)^{k+1}\right)-(-1)^{k+1}\cdot 11
$$
Alternate alternate solution When $k=2l$ is even then
$$
10^{2l}-1=100^l-1=(100-1)( \mbox{ junk }) 
$$
When $k=2l+1$ is odd then
$$
10^{2l+1}+1=10^{2l+1}-10+11=10 (100^l-1) +11=10 (100-1)( \mbox{ junk })  +11
$$
