In this section, we have studied two formulations of the Principle of Mathematical Inducti...

In this section, we have studied two formulations of the Principle of Mathematical Induction.

(a) Use either of these to establish the following (peculiar?) third formulation.

Suppose P(n) is a statement about the natural number n such that

1. p(1)−is true;

2. For any k > 1. P(k) true implies P(2k)true; and

3. For any k ≥ 2, P(k) true implies P(k − 1) true.

(b) Prove that, for any two nonnegative numbers x and

(c) Use the Principle of Mathematical Induction in the form given in part (a) to generalize the result of part (b), thus establishing the arithmetic mean-geometric mean inequality: For any n ≥ 1 and any n nonnegative real numbers a1, a2,…, an.