Answer
The property was proved for $n=1$
The property is correct if n is changed by $n+1$
Work Step by Step
Let's prove the property for $n=1$:
$5^n-1=5^1-1=4$
4 is divisible by 4. It is correct!
Suppose that the propety is correct, that is:
$5^n-1$ is divisible by 4 for all natural numbers.
Now, let's prove the property for $n+1$:
$5^{n+1}-1=5·5^n-1=(4+1)5^n-1=4·5^n+(5^n-1)$
Both terms are divisible by 4. So, the sum is also divisible by 4.
