Chapter 7: Q. 51 (page 592)

In Exercises 51–54 use the difference test in Theorem 7.6 to analyze the monotonicity of the given sequence.
$\{k^{2} - 5 k\}$

Short Answer

Expert verified

The given sequence is strictly increasing for $k \geq 2 .$

Step by step solution

Step 1. Given Information.

The given sequence is $\{k^{2} - 5 k\} .$

Step 2. Use the difference test.

To analyze the monotonicity of the given sequence we will use the difference test.

Let the general term of the sequence is $a_{k} = k^{2} - 5 k .$

So, the term $a_{k + 1}$ is

role="math" localid="1649229170271" $a_{k + 1} = {(k + 1)}^{2} - 5 (k + 1) a_{k + 1} = k^{2} + 2 k + 1 - 5 k - 5 a_{k + 1} = k^{2} - 3 k - 4$

According to the test,

role="math" localid="1649229456322" $a_{k + 1} - a_{k} \geq 0 k^{2} - 3 k - 4 - (k^{2} - 5 k) \geq 0 k^{2} - 3 k - 4 - k^{2} + 5 k \geq 0 2 k - 4 \geq 0 k \geq 2$

Thus, the sequence is strictly increasing for $k \geq 2 .$

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

In Exercises 51–54 use the difference test in Theorem 7.6 to analyze the monotonicity of the given sequence.
$\{k^{2} - 5 k\}$

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Use the difference test.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Logic and Functions

Decision Maths

Discrete Mathematics

Probability and Statistics

Applied Mathematics

Study anywhere. Anytime. Across all devices.

In Exercises 51–54 use the difference test in Theorem 7.6 to analyze the monotonicity of the given sequence.k2-5k

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Use the difference test.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Logic and Functions

Decision Maths

Discrete Mathematics

Probability and Statistics

Applied Mathematics

Study anywhere. Anytime. Across all devices.

In Exercises 51–54 use the difference test in Theorem 7.6 to analyze the monotonicity of the given sequence.
$\{k^{2} - 5 k\}$