<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] First note that $5^k-5$ is divisible by $4$ for all positive integers $k$.
Let $P(n)$ be the proposition that $2\times7^n+3\times5^n-5$ is a multiple of $24$.
$P(1)$ is the proposition that $2\times7+3\times5-5$ is a multiple of $24$ which is true
Suppose $P(k)$ is true and so $2\times7^k+3\times5^k-5$ is  a multiple of $24$.
Then $2\times7^{k+1}+3\times5^{k+1}-5=14\times 7^k+15\times5^k-5=7\times (2\times7^k+3\times5^k-5)-6\times(5^k-5)$
is divisible by $24$. Therefore $P(k+1)$ is true and we are finished.
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