<s>[INST] How to proof $\forall n>0: 5|(6^n-1)$ is true using mathematical induction This is what i did so far, i got stuck on the later steps of using mathematical induction
I started with a base of $1$ which gave me $5|5$ which is $1$. Then I used $n = k$, and i got $5|(6^k-1)$ and i assumed this was true, but when i got to $n = k+1$ i didnt knew how to implement $5|(6^k-1)$ to $5|(6^{k+1}-1)$ to show that the theorem is true. For the real assignment, check this link: https://imgur.com/e6Z4tZW
 [/INST] With induction. If $5|6^n-1$, then there is $q_n$ such that $6^n-1=5\cdot q_n$. Note that
$$
6^{n+1}-1=6\cdot 6^n-1=5\cdot 6^n+6^n-1=5(6^n+q_n)
$$
Without induction.
\begin{align}
6^n-1 
=&
(5+1)^n-1
\\
=&
(5^{n}+n\cdot 5^{n-1}+\frac{n(n-1)}{2}\cdot 5^{n-2}+\cdots+\frac{n(n-1)}{2}\cdot 5^2+ n\cdot5+1)-1
\\
=&
5^{n}+n\cdot 5^{n-1}+\frac{n(n-1)}{2}\cdot 5^{n-2}+\cdots+\frac{n(n-1)}{2}\cdot 5^2+ n\cdot5
\\
=&
5\cdot (5^{n-1}+n\cdot 5^{n-2}+\frac{n(n-1)}{2}\cdot 5^{n-3}+\cdots+\frac{n(n-1)}{2}\cdot 5+ n)
\\
\end{align}
</s>