   Chapter 2.2, Problem 47E

Chapter
Section
Textbook Problem

Use generalized induction and Exercise 43 to prove that n 2 < 2 n for all integers n ≥ 5 .(In connection with this result, see the discussion of counterexamples in the Appendix.) 1 + 2 n < 2 n for all integers n ≥ 3

To determine

To prove: n2<2n for all integers n5 by using generalized induction.

Explanation

Given information:

The given statement is, “ n2<2n for all integers n5

Formula used:

1) Strategy: Proof by Generalized Induction

1. Basic Step: The statement is verified for n=a.

2. Induction Hypothesis: The statement is assumed true for n=k, where ka.

3. Inductive Step: With this assumption made, the statement is then proved to be true for n=k+1.

2) 1+2n<2n for all integers n3.

Proof:

Consider the statement, “ n2<2n for all integers n5

For n=5

Taking L.H.S.,

52=25

Taking R.H.S,

25=32

25<32

Therefore, the statement is true for n=5.

Assume that the statement is true for n=k.

k2<2k where k5

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