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Elements Of Modern Algebra

8th Edition
Gilbert + 2 others
ISBN: 9781285463230
BuyFindarrow_forward

Elements Of Modern Algebra

8th Edition
Gilbert + 2 others
ISBN: 9781285463230
Textbook Problem

In Exercises 35 48 , use mathematical induction to prove that the given statement is true for all positive integers n .

3 is a factor of n 3 + 2 n .

To determine

To prove: 3 is a factor of n3+2n for all positive integers n.

Explanation

Formula used:

Mathematical Induction:

The given statement Pn is true for all positive integers n if,

a. Pn is true for n=1

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

Proof:

Consider the statement 3 is a factor of n3+2n for all positive integers n.

By mathematical induction,

a. For n=1,

As, n3+2n=13+21=3.

Since, 3 is a factor of 3.

Therefore, the statement is true for n=1.

b. Assume that the statement is true for n=k.

That is, 3 is a factor of k3+2k.

k3+2k=3z, for some integer z.

To show the statement is true for n=k+1

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