Chapter 12: Q 18. (page 830)

In Problem, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers $n$ .
$1 \cdot 2 + 3 \cdot 4 + 5.6 + . . . + (2 n - 1) (2 n) = \frac{1}{3} n (n + 1) ((4 n - 1)$

Short Answer

Expert verified

The statement is shown.

Step by step solution

Step 1. Given information

The statement is $1 \cdot 2 + 3 \cdot 4 + 5.6 + . . . + (2 n - 1) (2 n) = \frac{1}{3} n (n + 1) ((4 n - 1)$ .

Step 2. Show the given statement.

Put $n = 1$ in the statement $1 \cdot 2 + 3 \cdot 4 + 5.6 + . . . + (2 n - 1) (2 n) = \frac{1}{3} n (n + 1) ((4 n - 1)$

width="177">1.2=13,1.(1+1)(4.1-1)

So, the statement is true for $n = 1$ .

Next, we assume that the above statement is true for some $k$ . so that

width="359">1.2+3.4+5.6+⋯+(2k-1)(2k)=13k(k+1)(4k-1)

Now we consider the sum of first $k + 1$ terms.

$\begin{array}{rcl} 1.2 + 3.4 + 5.6 + \dots + (2 k - 1) (2 k) + (2 (k + 1) - 1) (2 (k + 1)) & = & \frac{1}{3} k (k + 1) (4 k - 1) + (2 k + 2 - 1) (2 (k + 1)) \\ = & \frac{(k + 1)}{3} (k (4 k + 3) + 2 (4 k + 3)) \\ = & \frac{(k + 1)}{3} (4 k^{2} + 3 k + 8 k + 6) \\ = & \frac{1}{3} (k + 1) ((k + 1) + 1) (4 (k + 1) - 1) \end{array}$

As a result, 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 $n$ .
$1 \cdot 2 + 3 \cdot 4 + 5.6 + . . . + (2 n - 1) (2 n) = \frac{1}{3} n (n + 1) ((4 n - 1)$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Show the given statement.

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.1·2+3·4+5.6+...+(2n-1)(2n)=13n(n+1)((4n-1)

Short Answer

Step by step solution

Step 1. Given information

Step 2. Show the given statement.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Theoretical and Mathematical Physics

Statistics

Logic and Functions

Discrete Mathematics

Pure Maths

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 $n$ .
$1 \cdot 2 + 3 \cdot 4 + 5.6 + . . . + (2 n - 1) (2 n) = \frac{1}{3} n (n + 1) ((4 n - 1)$