<s>[INST] Prove $7^{2n}-1$ is divisible by 48 Trying to prove $7^{2n}-1$ is divisible by 48.
I began by taking the base case $n=1$ so $7^{2}-1=48$ which is true.
Now from here taking the inductive step to get
$7^{2(n+1)}-1=48k$
$7^{2n+2}-1=48k$
$7^{2}*7^{2n}-1=48k$
Not quite sure where to go from here. Thanks.
 [/INST] Let S = $1 + 49 + 49^2 + \cdots + 49^{n-1} $
\begin{align}
        S &= 1 + 49 + 49^2 + \cdots + 49^{n-1} + 49^n - 49^n\\
        S &= 1 + 49(1 + 49 + 49^2 + \cdots + 49^{n-1}) - 49^n \\
        S &= 1 + 49S - 49^n \\
 49^n - 1 &= 48S\\
       48 &\mid 49^n - 1 \\
       48 &\mid 7^{2n} - 1 \\
\end{align}
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