   Chapter 2.2, Problem 34E

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 4 .       1 + 2 n ≤ 3 n

To determine

To prove: The statement 1+2n3n is true for all positive integers n by using mathematical induction.

Explanation

Given information:

The given statement is 1+2n3n.

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, “ 1+2n3n.”

For n=1, P1 is 1+2(1)=1+233

Therefore, P1 is true.

Assume that Pk is true.

1+2k3k

For n=k+1

The left side is 1+2(k+1)

By using distributive law,

1+2

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