Slight error with your induction step - if n = k, then . I think you replaced k with k+1 or something. Also, n = 2 yields 168.

Assuming the n = k case is true, you can show that n = k+1 works by showing that

, and proceed from there.

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A simpler solution would be to write and . Then our expression is equivalent to

Clearly, if n is odd or n is even, by substitution we obtain 0 mod 7, and we're done.