Chapter 12: Q. 9 (page 830)

In Problems 1–22, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers $n .$
$1 + 4 + 4^{2} + . . . + 4^{n - 1} = \frac{1}{3} (4^{n} - 1)$

Short Answer

Expert verified

The statement $1 + 4 + 4^{2} + . . . + 4^{n - 1} = \frac{1}{3} (4^{n} - 1)$ is true for all natural numbers.

Step by step solution

Step 1. Given information

The given statement is $1 + 4 + 4^{2} + . . . + 4^{n - 1} = \frac{1}{3} (4^{n} - 1) .$ We need to prove that the given statement is true for all natural numbers by using mathematical induction.

Step 2. Proof

Let's prove that the statement is true for $n = 1 .$

$4^{1 - 1} = \frac{1}{3} (4^{1} - 1) .$

$4^{0} = \frac{1}{3} (3) .$

$1 = 1 .$

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

Let's assume that the statement is true for $n = k .$

So, role="math" localid="1648054611339" $1 + 4 + 4^{2} + . . . + 4^{k - 1} = \frac{1}{3} (4^{k} - 1) .$

Let's prove that the statement is true for $n = k + 1 .$

$1 + 4 + 4^{2} + . . . + 4^{k - 1} + 4^{k + 1 - 1} .$

$= \frac{1}{3} (4^{k} - 1) + 4 k .$

$= \frac{4}{3} \cdot 4 k - \frac{1}{3} .$

$= \frac{1}{3} (4^{k + 1} - 1) .$

The statement is true for $n = k + 1 .$

Since the given statement is true for $n = 1$ and if the statement is true for some $n = k$ and it is also true for the next natural number $k + 1 .$

So, the statement is true for all natural numbers.

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 Problems 1–22, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers $n .$
$1 + 4 + 4^{2} + . . . + 4^{n - 1} = \frac{1}{3} (4^{n} - 1)$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Proof

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Mechanics Maths

Geometry

Discrete Mathematics

Decision Maths

Applied Mathematics

Study anywhere. Anytime. Across all devices.

In Problems 1–22, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers n.1+4+42+...+4n-1=134n-1

Short Answer

Step by step solution

Step 1. Given information

Step 2. Proof

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Mechanics Maths

Geometry

Discrete Mathematics

Decision Maths

Applied Mathematics

Study anywhere. Anytime. Across all devices.

In Problems 1–22, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers $n .$
$1 + 4 + 4^{2} + . . . + 4^{n - 1} = \frac{1}{3} (4^{n} - 1)$