Prove by mathematical induction that for every integer n that is greater than or equal to zero, 5 divides 6n+1 − 1.

Question

Can someone show a step by step process on how to solve this problem.

Answer 1

No matter what, the statement is false in the present form (independent of possibly forgotten brackets) and in particular not provable by induction.

$5\not\mid 6\cdot 1 +1-1=6$

$5\not\mid 6(1 +1)-1=11$

$5\not\mid 6\cdot 1 +(1-1)=6$

Answer 2

If you meant $5$ divides $6^{n+1}-1$, here's an outline for a proof by induction.

Base case: $n=0$: $5$ divides $6^1-1=5$.

Inductive step: Assume $5$ divides $6^{n}-1$.

Then can you see how to show that $5$ divides $6^{n+1}-1=6\times6^n-1=5\times6^n+6^n-1$?