Prove by Induction of a Total Function I'm given a total function that spans Positive Integers ($\mathbb Z^+$ to Positive Real numbers $\mathbb R^+$)
I'm given that $S(1) = 7$
Also I'm given that
$S(2^k) = S(2^{k-1}) +5$ // equivalent to $2^{k-1} + 5$ (Right hand side)
where $k$ is an element of positive Integers
I'm told to prove that :
$S(2^k) = 5k+7$
I started doing the Bases step:

Since $S(1) = 7$, LHS when $k=0$ $= S(1)$, RHS when $K = 0 = 5(0) + 7 = 7$
Thus, $S(1)$ Holds

Then I did the Induction Hypothesis:
Assume true for $S(n)$
$S(2^n) = 5n+7$
But I'm not sure how to do the inductive step, I have:
Try for $n = k+1$
$S(2^{k+1}) = 5(k+1) +7$ //left hand side reads $2^{k+1}$
Any help would be much appreciated.

You need to prove it for $n+1$ assuming that it's true for $n$.
$$S(2^{n+1})=S(2^{n})+5=5n+7+5=5(n+1)+7$$
Where I have used the given condition in the first equality and the induction hypothesis in the second one. Doing this you have proved the given formula using mathematical induction.
