Chapter 12: Q 27. (page 830)

In Problem, prove each statement.
${(1 + a)}^{n} \geq 1 + n a$ for $a > 0$

Short Answer

Expert verified

The given statement is shown.

Step by step solution

Step 1. Given information

${(1 + a)}^{n} \geq 1 + n a$

Step 2. Prove the given statement.

Put $n = 1$ in $(1 + a)^{n} \geq 1 + n a$ to check if the statement is true for $n = 1$ .

width="121">(1+a)1≥1+(1)a1+a≥1+a

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

Assume that the statement is true for width="37">n=k

Thus, $(1 + a)^{k} \geq 1 + k a$ .

Find whether $(1 + a)^{k + 1} \geq 1 + (k + 1) a$

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

Thus, the statement is true for $k + 1$

Since both the conditions of the Principle of Mathematical Induction are satisfied, the statement

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 Problem, prove each statement.
${(1 + a)}^{n} \geq 1 + n a$ for $a > 0$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Prove the given statement.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Probability and Statistics

Pure Maths

Theoretical and Mathematical Physics

Applied Mathematics

Geometry

Study anywhere. Anytime. Across all devices.

In Problem, prove each statement.1+an≥1+nafora>0

Short Answer

Step by step solution

Step 1. Given information

Step 2. Prove the given statement.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Probability and Statistics

Pure Maths

Theoretical and Mathematical Physics

Applied Mathematics

Geometry

Study anywhere. Anytime. Across all devices.

In Problem, prove each statement.
${(1 + a)}^{n} \geq 1 + n a$ for $a > 0$