Using the summation formula in an inductive proof

Question

I'm not sure what to do. My instructors feedback is that I cannot use the geometric sum in the proof because I have to use induction. I don't understand this feedback because I don't see how the two are mutually exclusive. I also think that I am using induction. The summation formula for the geometric sum only proves the inductive step and not the original equation. There's also some corrections I need to make to my base case, so please forgive me. But can you all help me with the final inductive step? my proof instructors feedback

Answer 1

I agree with your instructor, you cannot use geometric series, because that is effectively what you are trying to prove. The base case looks good. The inductive step would go something like this:

Assume the equation holds for $n=k.$ Then, we have that $$3\sum_{i=1}^{k-1}4^i=4^k-4.$$ Now, take $n=k+1$ on the LHS. We have that $$3\sum_{i=1}^{k}4^i=3\sum_{i=1}^{k-1}4^i+(3\cdot4^k)=4^k-4+(3\cdot4^k)=4\cdot 4^k-4=4^{k+1}-4.$$ Thus, we see that if equality holds for $n=k,$ then equality holds for $n=k+1$ and completed our inductive step. In conjunction with the base case, this completes the proof.