Proving by induction is sometimes bad

Today in my numerical analysis homework I encounter this problem,

I think questions like this are bad. Because I think in many real world situations, I won’t have the formula that you could do induction on.
So for that homework problem, I decided to derive the formula from scratch and I did solve it after 2 hours.
For the easier case, addition/substraction, I expand a few terms from the linear recurrence function(f(n) = n-1 + f(n-1)).
For multiplication/division, I have to expand a few example terms(i.e expanding f(5)) from function g(n) = n*g(n-1) + n to find out the pattern.

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