<s>[INST] Proof by Induction $2\cdot 7^n + 3\cdot 5^n - 5$ is multiple of $24$ Proof by induction that $2\cdot 7^n + 3\cdot 5^n - 5$ is a multiple of $24$.
 I tried solving but got stuck. 
Show that it is true for n=1
 $$2\cdot 7^1 + 3\cdot 5^1 - 5 = 14 + 15 - 5 = 24$$
Assume it true for $n = k$
$$2\cdot 7^k + 3\cdot 5^k - 5 = 24g$$
Show it is true for $n= k + 1$
$$2\cdot 7^{k+1} + 3\cdot 5^{k+1} - 5$$
 is a multiple of 24
$$(2\cdot 7)(2\cdot 7^k) + (3\cdot 5^k)(3\cdot 5) - 5$$
$$2\cdot7(24g + 5)3\cdot 5 - 5$$
 I'm stuck and don't know how to proceed
 [/INST] Notice that $5^2$ and $7^2$ are both one more than a multiple of $24.$  Prove the cases $k=1$ and $k=2$ for your basis step. 
Then in the induction you have, for one term,
$$2(7^{k+2}) = 2(7^k)(48+1).$$
Do the same for the other term.  Induct by twos instead of by ones.
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