Chapter 12: Q 17. (page 830)

In Problem, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers $n$ .
role="math" localid="1647675344162" $1 \cdot 2 + 2 \cdot 3 + 3 \cdot 4 + . . . + n (n + 1) = \frac{1}{3} n (n + 1) (n + 2)$

Short Answer

Expert verified

The given statement has been shown.

Step by step solution

Step 1. Given information

Statement is $1 \cdot 2 + 2 \cdot 3 + 3 \cdot 4 + . . . + n (n + 1) = \frac{1}{3} n (n + 1) (n + 2)$

Step 2. Show the given statement using the principle of mathematical induction.

For $n = 1$

$\begin{array}{rcl} \frac{1}{3} (1) (1 + 1) (1 + 2) & = & \frac{1}{3} (2) (3) \\ = & 2 \end{array}$

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

$1 \cdot 2 + 2 \cdot 3 + 3 \cdot 4 + \dots \dots \dots \dots + k (k + 1) = \frac{1}{3} k (k + 1) (k + 2)$

style="max-width: none; vertical-align: -89px;" width="552" $\begin{array}{rcl} \frac{1}{3} (k + 1) ((k + 1) + 1) ((k + 1) + 2) & = & 1 \cdot 2 + 2 \cdot 3 + \dots \dots \dots + k (k + 1) + (k + 1) ((k + 1) + 1) \\ = & \frac{1}{3} k (k + 1) (k + 2) + (k + 1) (k + 2) \\ = & \frac{1}{3} (k + 1) (k + 1 + 1) [k + 1 + 2] \\ = & \frac{1}{3} (k + 1) [(k + 1) + 1] [(k + 1) + 2] \end{array}$

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.

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In Problem, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers $n$ .
role="math" localid="1647675344162" $1 \cdot 2 + 2 \cdot 3 + 3 \cdot 4 + . . . + n (n + 1) = \frac{1}{3} n (n + 1) (n + 2)$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Show the given statement using the principle of mathematical induction.

One App. One Place for Learning.

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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.role="math" localid="1647675344162" 1·2+2·3+3·4+...+n(n+1)=13n(n+1)(n+2)

Short Answer

Step by step solution

Step 1. Given information

Step 2. Show the given statement using the principle of mathematical induction.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Applied Mathematics

Statistics

Theoretical and Mathematical Physics

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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 $n$ .
role="math" localid="1647675344162" $1 \cdot 2 + 2 \cdot 3 + 3 \cdot 4 + . . . + n (n + 1) = \frac{1}{3} n (n + 1) (n + 2)$