Is it true that $13|2^n5^{2n+1}+3^{2n+1}7^{n+1}$ for all $n$? Is it true that $13|2^n5^{2n+1}+3^{2n+1}7^{n+1}$ for all $n$?
So what I did was basically simplify the terms on the right $\mod 13$.
$2^n5^{2n+1}+3^{2n+1}7^{n+1} \mod 13$
$= 5\cdot 2^n\cdot 5^n\cdot 5^n+3\cdot 7\cdot 3^{n}\cdot 3^n7^n \mod 13$
$= 5\cdot 50^n+21\cdot 63^n \mod 13$
$= 5\cdot 50^n+21\cdot 63^n \mod 13$
$= 5\cdot 11^n+8\cdot 11^n \mod 13$
$= 13\cdot 11^n \mod 13$
$= 0\cdot 11^n \mod13$
$=0 \mod13$
Thus, for all $n$, $13$ will divide this term, and the statement is true. Can anyone tell me if what I have done is correct/wrong? Would really appreciate it!

Yes it is correct and very nice. The only minor note, we can also conclude  at that step of course $$...\equiv 13\cdot 11^n \mod 13$$
As an alternative you can try also by induction.
