Let P(n) be the statement that n! < nn where n is an integer greater than 1.

Question

a) What is the statement P(2)?
b) Show that P(2) is true, completing the basis step of the proof.
c) What is the inductive hypothesis?
d) What do you need to prove in the inductive step?
e) Complete the inductive step.
f) Explain why these steps show that this formula is true when ever is an integer greater than 1.

Answer

a) Statement P(2):

P(2) states that 2! < 2².

b) Basis step:

2! = 2 * 1 = 2 2² = 2 * 2 = 4 2 < 4 Therefore, P(2) is true.

c) Inductive hypothesis:

Assume P(k) is true for some integer k > 1, meaning k! < k^k.

d) Inductive step:

We need to prove that P(k + 1) is true, meaning (k + 1)! < (k + 1)^(k + 1).

e) Completing the inductive step:

(k + 1)! = (k + 1) * k! < (k + 1) * k^k (using the inductive hypothesis) < (k + 1) * (k + 1)^k (since k + 1 > k) = (k + 1)^(k + 1) Therefore, P(k + 1) is true.

f) Explanation:

  • The basis step establishes the truth of P(2), providing a starting point for the induction.
  • The inductive hypothesis assumes the truth of P(k) for a general integer k > 1.
  • The inductive step demonstrates that if P(k) is true, then P(k + 1) must also be true.
  • This chain of implications, starting from P(2) and extending to all subsequent integers, proves the validity of P(n) for all integers n > 1.

This process of mathematical induction guarantees that the inequality n! < n^n holds for all integers greater than 1.

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