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Chapter 12: Q 19. (page 830)

In Problem, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers n.

n2+nis divisible by2.

Short Answer

Expert verified

The statement is shown.

Step by step solution

01

Step 1. Given information

The expression isn2+n.

02

Step 2. Show that the statement is true for all natural numbers.

Substitute n=1in n2+n.

(1)2+1=2which is divisible by 2.

Thus, the statement is true for n=1

Assume that the statement is true for n=k

Thus, k2+kis divisible by 2

(k+1)2+(k+1)=k2+2k+1+k+1=k2+k+2k+2=k2+k+2(k+1)

Since k2+kis divisible by 2 and 2(k+1)is also divisible by 2,(k+1)2+(k+1)is divisible by 2 .

Thus, the statement is true for k+1.

Since both the conditions of the Principle of Mathematical Induction are satisfied, the statement is true for all natural numbers.

Therefore, n2+nis divisible by 2 .

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