I'm not sure how to conclude what the Inductive Hypothesis for this Strong/Complete Induction proof

Question

I checked the base case and it worked, but I'm not really sure what the hypothesis should be?

$$n = k, a_k < (\frac{5}{3})^k$$

If we want to prove $$k+1$$ then we have this as what we're trying to proof: $$P(k+1) = a_{k+1} < (\frac{5}{3})^{k+1}$$

Answer 1

HINT

We have that

$$a_{k+1} = a_k + a_{k-1}\stackrel{Ind. Hyp.}<\left(\frac{5}{3}\right)^{k}+\left(\frac{5}{3}\right)^{k-1} =\left(\frac{5}{3}\right)^{k}\left(1+\left(\frac{5}{3}\right)^{-1}\right)\stackrel{?}<\left(\frac{5}{3}\right)^{k+1}$$

and the latter requires

$$\left(\frac{5}{3}\right)^{k}\left(1+\left(\frac{3}{5}\right)\right)\stackrel{?}<\left(\frac{5}{3}\right)^{k}\left(\frac{5}{3}\right)$$