# proof by induction 5^n-1 is divisible by 4

• Nov 28th 2011, 07:25 PM
Jskid
proof by induction 5^n-1 is divisible by 4
Use mathematical induction to prove that the assertion is true for $n\ge 1$. $5^n-1$ is divisible by $4$.

basis $n_{o}=1$ then $5^1-1=4$ is divisible by $4$.

IH let $1\le l and assume $5^l-1$ is divisible by $4$.

Induction Step
$5^{k}-1=(5)5^{k-1}-1$
Since $5^{k-1}-1$ is divisble by $4$ (by the induction hypothesis) then $(5)5^{k-1}-1$ is also divisible by $4$

Is this right?
What is wrong with the latex
• Nov 28th 2011, 07:32 PM
pickslides
Re: proof by induction 5^n-1 is divisible by 4
The induction step should have n = k+1
• Nov 29th 2011, 10:54 AM
Annatala
Re: proof by induction 5^n-1 is divisible by 4
Quote:

IH let $1\le l and assume $5^l-1$ is divisible by $4$.
If this is a proof by induction you need to go a step at a time, not prove it for all numbers at once.

I'd say this. "Assume $4 |(5^k-1)$. We want to show $4|( 5^k^+^1-1)$."

Quote:

Since $5^{k-1}-1$ is divisble by $4$ (by the induction hypothesis) then $(5)5^{k-1}-1$ is also divisible by $4$
Not quite, no. You have the formula wrong, and you need to show your work; how do you know 4 divides $5^k^+^1-1$ for certain?