Is the following Strong Induction conclusion true?

Question

can someone verify if this is true?

I assume that P(i) holds for i < n, but then I would believe that this means the conclusion should be 3^n-1 and not 3^n.

However, the correct answer given by my teacher says 3^n is correct.

Answer 1

By assuming $P(i)$ holds for $i<n$, clearly, we have $f(n-1)=3^{n-1}$ from this assumption.

Our goal for the induction step is to argue that from properties $$P(i) \text{ holds for } i<n\tag{1}$$ and $$f(n)=4\cdot f(n-1)-3f(n-2)\tag{2},$$ we can conclude $P(i)$ holds for $i=n$, that is $f(n)=3^n$.