   Chapter 2.2, Problem 36E

Chapter
Section
Textbook Problem

In Exercise 32 − 36 use mathematical induction to prove that the given statement is true for all positive integers n . 3 6 .           n ! ≤     n n

To determine

To prove: The statement n!nn is true for all positive integers n by using mathematical induction.

Explanation

Given information:

The given statement is n!nn.

Formula used:

For all positive integers n, the statement Pn is true if,

a. Pn is true for n=1

b. The truth of Pk always implies that Pk+1 is true.

Proof:

Let Pn be the statement, “ n!nn.”

For n=1,

Taking L.H.S,

1!=1

Taking R.H.S,

11=1

Therefore, P1 is true.

Assume that Pk is true.

k!kk

For n=k+1

The left side is

(k+1)!=123k(k+1)

This can be written as

=[123k

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