Question

3power(2n+2) - 9 is divisible by 72

Answer 1

$let's\ prove\ it\ by\ induction. \\ \\ for\ n=1,\\ =3^{2*1+2}-9=81-9=72 \\ since\ 72\ is \ divisible\ by\ 72,\ it\ is\ true\ for\ n=1. \\ \\ let\ is\ true\ for\ n=k.\\So\ 3^{2k+2}-9\ is\ divisible\ by\ 72\\or\ 3^{2k+2}-9=72m\ where\ m\ is\ a\ natural\ number.\\ \\we\ need\ to\ prove\ that\ it\ is\ true\ for\ (k+1).\\ \\ 3^{2(k+1)+2}-9\\ =3^{2k+2+2}-9\\ =3^{2k+2}*3^2-9\\ =9*3^{2k+2}-9\\ = 9*3^{2k+2}-9-72+72\\ =9*3^{2k+2}-81+72\\ =9*(3^{2k+2}-9)+72\\ =9*(72m)+72\\ =72*(9m+1)$
$since\ 72*(9m+1)\ is\ always\ divisible\ by\ 72,\\ 3^{2n+2}-9\ is\ always\ divisible\ by\ 72.\ (proved)$