   Chapter 2.3, Problem 40E

Chapter
Section
Textbook Problem

In Exercises 35 − 48 , use mathematical induction to prove that the given statement is true for all positive integers n . 6 is a factor of n 3 + 5 n .

To determine

To prove: 6 is a factor of n3+5n for all positive integers n.

Explanation

Formula used:

i) 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.

ii) For any integer a, 2|a(a+1).

Proof:

Consider the statement 6 is a factor of n3+5n for all positive integers n.

By mathematical induction,

a. For n=1,

As, n3+5n=13+51=1+5=6.

Since, 6 is a factor of 6.

Thus, 6 is a factor of n3+5n for n=1.

Therefore, the statement is true for n=1.

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

That is, 6 is a factor of k3+5k.

k3+5k=6z, for some integer z.

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

Now,

(k+1)3+5(k+1)

=k3+3k+3k2+1+5k+5

=(k3+5k)+<

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