Algebra 2
Algebra 2
1st Edition
ISBN: 9780078884825
Author: McGraw-Hill/Glencoe
Publisher: Glencoe/McGraw-Hill School Pub Co
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Chapter 11.7, Problem 5CYU
To determine

To prove that 4n1 is divisible by 3 for all natural numbers n.

Expert Solution & Answer
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Explanation of Solution

Given:

  4n1

Concept Used:

An identity can be proved using mathematical induction in two steps:

  1. First prove the identity is true for n=1 .
  2. Prove that if it is true for n=k then it is also true for n=k+1 .

Calculation:

In order to prove that 4n1 is divisible by 3 for all natural numbers n, use mathematical induction.

First, let’s check if this is true for n=1 .

For n=1

  411=41=3

And 3 is divisible by 3. Thus, the given identity is true for n=1 .

Now, let it is true for n=k; for some k>1 .

Thus, (4k1) is divisible by 3, i.e.

  4k1=3m , for some integer m .

Now, let’s show that it is also true for n=k+1 , i.e. prove that (4k+11) is also divisible by 3. Since,

  4k+11=4k41=(4k1+1)41=(3m+1)41  4k1=3m=3m4+41=3m4+3=3(4m+1)

Thus, (4k+11) is a multiple of 3, i.e. divisible by 3, i.e. the identity is also true for n=k+1 , when it is true for n=k .

Thus, by mathematical induction it is proved that 4n1 is divisible by 3 for all natural numbers n .

Chapter 11 Solutions

Algebra 2

Ch. 11.1 - Prob. 1CYUCh. 11.1 - Prob. 2CYUCh. 11.1 - Prob. 3CYUCh. 11.1 - Prob. 4CYUCh. 11.1 - Prob. 5CYUCh. 11.1 - Prob. 6CYUCh. 11.1 - Prob. 7CYUCh. 11.1 - Prob. 8CYUCh. 11.1 - Prob. 9CYUCh. 11.1 - Prob. 10CYUCh. 11.1 - Prob. 11CYUCh. 11.1 - Prob. 12CYUCh. 11.1 - Prob. 13CYUCh. 11.1 - Prob. 14CYUCh. 11.1 - Prob. 15CYUCh. 11.1 - Prob. 16CYUCh. 11.1 - Prob. 17CYUCh. 11.1 - Prob. 18CYUCh. 11.1 - Prob. 19CYUCh. 11.1 - Prob. 20CYUCh. 11.1 - Prob. 21PPSCh. 11.1 - Prob. 22PPSCh. 11.1 - Prob. 23PPSCh. 11.1 - Prob. 24PPSCh. 11.1 - Prob. 25PPSCh. 11.1 - Prob. 26PPSCh. 11.1 - Prob. 27PPSCh. 11.1 - Prob. 28PPSCh. 11.1 - Prob. 29PPSCh. 11.1 - Prob. 30PPSCh. 11.1 - Prob. 31PPSCh. 11.1 - Prob. 32PPSCh. 11.1 - Prob. 33PPSCh. 11.1 - Prob. 34PPSCh. 11.1 - Prob. 35PPSCh. 11.1 - Prob. 36PPSCh. 11.1 - Prob. 37PPSCh. 11.1 - Prob. 38PPSCh. 11.1 - Prob. 39PPSCh. 11.1 - Prob. 40PPSCh. 11.1 - Prob. 41PPSCh. 11.1 - Prob. 42PPSCh. 11.1 - Prob. 43PPSCh. 11.1 - Prob. 44PPSCh. 11.1 - Prob. 45PPSCh. 11.1 - Prob. 46PPSCh. 11.1 - 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Publisher:Cengage Learning
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ISBN:9780135163078
Author:Michael Sullivan
Publisher:PEARSON
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ISBN:9780980232776
Author:Gilbert Strang
Publisher:Wellesley-Cambridge Press
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ISBN:9780077836344
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What is a Relation? | Don't Memorise; Author: Don't Memorise;https://www.youtube.com/watch?v=hV1_wvsdJCE;License: Standard YouTube License, CC-BY
RELATIONS-DOMAIN, RANGE AND CO-DOMAIN (RELATIONS AND FUNCTIONS CBSE/ ISC MATHS); Author: Neha Agrawal Mathematically Inclined;https://www.youtube.com/watch?v=u4IQh46VoU4;License: Standard YouTube License, CC-BY