Prove the following statement SHOW ALL WORK BE AS DETAILED A

Prove the following statement.

SHOW ALL WORK. BE AS DETAILED AS POSSIBLE. WRITE NEATLY!

THIS WILL BE PROVEN BY INDUCTION. SHOW THE BASIS CASE AND THE INDUCTION STEP!

Solution

Base Case: n=1

10^1=10=11-1=0 -1 =-1 mod 11

Inductive Step: Assume true for some n>=1

We show it is true for n+1

10^{n+1}=10^n*10=(-1)^n*(11-1) mod 11=11*(-1)^n+(-1)^{n+1} mod 11=0+(-1)^{n+1} mod 11

Hence proved to be true for n+1

Hence true for all n>=1