Use mathematical induction to prove that $3\cdot 5^{2n+1} +2^{3n+1}$
is divisible by $17$ for all $n ∈ \mathbb{N}$.
I've tried to do it as follow.
If $n = 1$ then $392/17 = 23$.
Assume it is true when $n = p$. Therefore $3\cdot 5^{2p+1} +2^{3p+1} = 17k $ where $k ∈ \mathbb{N} $. Consider now $n=p+1$. Then
\begin{align}
&3\cdot 5^{2(p+1)+1} +2^{3(p+1)+1}=\\
&3\cdot 5^{2p+1+2} + 2^{3p+1+3}=\\
&3\cdot5^{2p+1}\cdot 5^{2} + 2^{3p+1}\cdot 2^{3}.
\end{align}
I reached a dead end from here. If someone could help me in the direction of the next step it would be really helpful. Thanks in advance.
Best Answer
Having shown that $$3\cdot5^{2p+1}+2^{3p+1}=17k$$ we have (continuing from the last line) $$3\cdot5^{2p+1}5^{2}+2^{3p+1}2^{3}=17(3\cdot5^{2p+1})+8(3\cdot5^{2p+1}+2^{3p+1})=17(3\cdot5^{2p+1}+8k)$$ so the next expression is also divisible by 17, as required.