Divisibility by induction

Question

I have learnt how to prove expressions by induction based on the use of three assumptions

1. $n=1$,
2. $n=k$,
3. $n=k+1$.

But can someone help me prove that $1+10^{2n-1}$ is divisible by $11$

Answer 1

With $n=1$, the expression becomes $1+10^{2\cdot 1-1}$. Is that divisible by $11$?

With $n=k$ and $n=k+1$, the expression becomes $1+10^{2k-1}$ and $1+10^{2k+1}$, respectively. Their difference is $10^{2k+1}-1-(10^{2k-1}-1)=10^{2k-1}\cdot(10^2-1)$, so if one is divisible by $11$, then ...?