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 $n$ .
$n^{2} + n$ is divisible by $2$ .

Short Answer

Expert verified

The statement is shown.

Step by step solution

Step 1. Given information

The expression is $n^{2} + n$ .

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

Substitute $n = 1$ in $n^{2} + n$ .

$(1)^{2} + 1 = 2$ which is divisible by $2$ .

Thus, the statement is true for $n = 1$

Assume that the statement is true for $n = k$

Thus, $k^{2} + k$ is divisible by $2$

$\begin{array}{rcl} (k + 1)^{2} + (k + 1) & = & k^{2} + 2 k + 1 + k + 1 \\ = & k^{2} + k + 2 k + 2 \\ = & (k^{2} + k) + 2 (k + 1) \end{array}$

Since $k^{2} + k$ is 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, $n^{2} + n$ is divisible by 2 .

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In Problem, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers $n$ .
$n^{2} + n$ is divisible by $2$ .

Short Answer

Step by step solution

Step 1. Given information

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

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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

Step by step solution

Step 1. Given information

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

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Calculus

Discrete Mathematics

Decision Maths

Pure Maths

Probability and Statistics

Study anywhere. Anytime. Across all devices.

In Problem, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers $n$ .
$n^{2} + n$ is divisible by $2$ .