<s>[INST] Prove by induction that $21 | 4^{n+1} + 5^{2n-1}$ The problem that I have is: Prove by induction that 21 divides 4n+1 + 52n-1
So far I have:
Base Case:
$n = 1$
$4^{1+1} + 5^{2-1} = 4^2 + 5^1 = 16 + 5 = 21$
Inductive Step:
Assume: $4^{k+1} + 5^{2k-1} = 3m$
$4^{(k+1)+1} + 5^{2(k+1)-1} = 3m$
$4^{k+2} + 5^{2k+1} = 3m$
I'm pretty sure I am far off on this one and not sure where to go. 
Thanks
 [/INST] Consider this:
$$4^{n+2} + 5^{2n+1} = 4\cdot 4^{n+1} + (21+4)\cdot 5^{2n-1} = 21\cdot5^{2n-1}  + 4\cdot (4^{n+1}+5^{2n-1})$$
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