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 $n=1$:
$2^{2n+1}+1=2^{2(1)+1}+1=2^5+1=32+1=33=3(11)$
3 is a factor. It is correct!
Suppose that the property is correct, that is:
$2^{2n+1}+1$ has 3 as a factor for all integers $n\geq1$
Now, let's prove the property for $n+1$:
$2^{2(n+1)+1}+1=2^{2n+3}+1=2^{2+2n+1}+1=2^2·2^{2n+1}+1=4·2^{2n+1}+1=(3+1)2^{2n+1}+1=3·2^{2n+1}+(2^{2n+1}+1)$
It is a sum of two terms that have 3 as a factor.
