Prove the following by math. induction. The format of the proof should be as follows:

1) State what, statement you are proving (it, should have a variable on which you do induction as a parameter).

2) State the base base(s) and prove them.

3) State the induction hypothesis

4) State and prove the induction step.

(a) Show limit for all n. 2 ?2, 5 ⁿ + 9 6 ⁿ .

(b) Show that .

(0) Consider a sequence defined as follows: s ₀ = l, s ₁ = 2 and for n > 2, s _n = l + max {s _[n/2] ,s _[n/2] }

(Recall that [x] is the door of x, that is, largest integer y x, and [x] is the ceiling of x, smallest integer y > x.)

Provo by strong induction that Vn.

and s _[n/2] . Which of them are the same and when? What is the relationship among ones that are not the same?

Vn0, the number 7+2 +82n+1 is divisible by 57 > 0, $n$n-1. Hint: compare s[(n+1)/2], S[(n+1)/2], [n/2]