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 $n=1$:
$2^{2n-1}+3^{2n-1}=2^{2(1)-1}+3^{2(1)-1}=2^1+3^1=5$
5 is a factor. It is correct!
Suppose that the property is correct, that is:
$2^{2n-1}+3^{2n-1}$ has 5 as a factor for all integers $n\geq1$
Now, let's prove the property for $n+1$:
$2^{2(n+1)-1}+3^{2(n+1)-1}=2^{2n+2-1}+3^{2n+2-1}=2^2·2^{2n-1}+3^2·3^{2n-1}=4·2^{2n-1}+(4+5)·3^{2n-1}=4(2^{2n-1}+3^{2n-1})+5·3^{2n-1}$
It is a sum of two terms that have 5 as a factor.
