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 1CYP
To determine

To prove

  12+22+32+........+n2=n(n+1)(2n+1)6

Expert Solution & Answer
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Explanation of Solution

Given:

  12+22+32+........+n2=n(n+1)(2n+1)6

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

  12+22+32+........+n2=n(n+1)(2n+1)6

Use mathematical induction.

First, let’s proof that the identity is true for n=1 .

The left hand side of given identity is:

  12=1

And right hand side is:

  1(n+1)(2(n)+1)6=1(1+1)(2(1)+1)6(Putting n=1)=2×36=1=Left side

Thus, the given identity is true for n=1 .

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

  12+22+32+........+k2=k(k+1)(2k+1)6

Now, let’s show that it is also true for n=k+1 , i.e. prove

  12+22+32+........+k2+(k+1)2=(k+1)(k+1+1)(2(k+1)+1)6

Take left side of above and simplify as

  12+22+32+........+k2+(k+1)2=k(k+1)(2k+1)6+(k+1)212+22+32+........+k2=k(k+1)(2k+1)612+22+32+........+k2+(k+1)2=k(k+1)(2k+1)+6(k+1)26=(k+1)(k(2k+1)+6(k+1))6=(k+1)(2k2+k+6k+6)6=(k+1)(2k2+7k+6)6=(k+1)(2k2+4k+3k+6)6=(k+1)(2k(k+2)+3(k+2))6=(k+1)(2k+3)(k+2)6=(k+1)(k+1+1)(2(k+1)+1)6=Right side

Thus, the identity is also true for n=k+1 , when it is true for n=k .

Thus by mathematical induction it is proved that

  12+22+32+........+n2=n(n+1)(2n+1)6

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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Prob. 50STPCh. 11.7 - Prob. 51SCh. 11.7 - Prob. 52SCh. 11.7 - Prob. 53SCh. 11.7 - Prob. 54SCh. 11.7 - Prob. 55SCh. 11.7 - Prob. 56SCh. 11.7 - Prob. 57SCh. 11.7 - Prob. 58SCh. 11.7 - Prob. 59SCh. 11.7 - Prob. 60SCh. 11.7 - Prob. 61SCh. 11.7 - Prob. 62SCh. 11 - Prob. 1QCCh. 11 - Prob. 2QCCh. 11 - Prob. 3QCCh. 11 - Prob. 4QCCh. 11 - Prob. 5QCCh. 11 - Prob. 6QCCh. 11 - Prob. 7QCCh. 11 - Prob. 8QCCh. 11 - Prob. 9QCCh. 11 - Prob. 10QCCh. 11 - Prob. 11QCCh. 11 - Prob. 12QCCh. 11 - Prob. 13QCCh. 11 - Prob. 14QCCh. 11 - Prob. 1MCQCh. 11 - Prob. 2MCQCh. 11 - Prob. 3MCQCh. 11 - Prob. 4MCQCh. 11 - Prob. 5MCQCh. 11 - Prob. 6MCQCh. 11 - Prob. 7MCQCh. 11 - Prob. 8MCQCh. 11 - Prob. 9MCQCh. 11 - Prob. 10MCQCh. 11 - Prob. 11MCQCh. 11 - Prob. 12MCQCh. 11 - Prob. 13MCQCh. 11 - Prob. 14MCQCh. 11 - Prob. 15MCQCh. 11 - Prob. 16MCQCh. 11 - Prob. 17MCQCh. 11 - Prob. 18MCQCh. 11 - Prob. 19MCQCh. 11 - Prob. 20MCQCh. 11 - Prob. 1SGRCh. 11 - Prob. 2SGRCh. 11 - Prob. 3SGRCh. 11 - Prob. 4SGRCh. 11 - Prob. 5SGRCh. 11 - Prob. 6SGRCh. 11 - Prob. 7SGRCh. 11 - Prob. 8SGRCh. 11 - Prob. 9SGRCh. 11 - Prob. 10SGRCh. 11 - Prob. 11SGRCh. 11 - Prob. 12SGRCh. 11 - Prob. 13SGRCh. 11 - Prob. 14SGRCh. 11 - Prob. 15SGRCh. 11 - Prob. 16SGRCh. 11 - Prob. 17SGRCh. 11 - Prob. 18SGRCh. 11 - Prob. 19SGRCh. 11 - Prob. 20SGRCh. 11 - Prob. 21SGRCh. 11 - Prob. 22SGRCh. 11 - Prob. 23SGRCh. 11 - Prob. 24SGRCh. 11 - Prob. 25SGRCh. 11 - Prob. 26SGRCh. 11 - Prob. 27SGRCh. 11 - Prob. 28SGRCh. 11 - Prob. 29SGRCh. 11 - Prob. 30SGRCh. 11 - Prob. 31SGRCh. 11 - Prob. 32SGRCh. 11 - Prob. 33SGRCh. 11 - Prob. 34SGRCh. 11 - Prob. 35SGRCh. 11 - Prob. 36SGRCh. 11 - Prob. 37SGRCh. 11 - Prob. 38SGRCh. 11 - Prob. 39SGRCh. 11 - Prob. 40SGRCh. 11 - Prob. 41SGRCh. 11 - Prob. 42SGRCh. 11 - Prob. 43SGRCh. 11 - Prob. 44SGRCh. 11 - Prob. 45SGRCh. 11 - Prob. 46SGRCh. 11 - Prob. 47SGRCh. 11 - Prob. 48SGRCh. 11 - Prob. 49SGRCh. 11 - Prob. 50SGRCh. 11 - Prob. 51SGRCh. 11 - Prob. 52SGRCh. 11 - Prob. 53SGRCh. 11 - Prob. 54SGRCh. 11 - Prob. 55SGRCh. 11 - Prob. 56SGRCh. 11 - Prob. 57SGRCh. 11 - Prob. 58SGRCh. 11 - Prob. 59SGRCh. 11 - Prob. 60SGRCh. 11 - Prob. 61SGRCh. 11 - Prob. 62SGRCh. 11 - Prob. 63SGRCh. 11 - Prob. 64SGRCh. 11 - Prob. 65SGRCh. 11 - Prob. 66SGRCh. 11 - Prob. 67SGRCh. 11 - Prob. 68SGRCh. 11 - Prob. 69SGRCh. 11 - Prob. 1PTCh. 11 - Prob. 2PTCh. 11 - Prob. 3PTCh. 11 - Prob. 4PTCh. 11 - Prob. 5PTCh. 11 - Prob. 6PTCh. 11 - Prob. 7PTCh. 11 - Prob. 8PTCh. 11 - Prob. 9PTCh. 11 - Prob. 10PTCh. 11 - Prob. 11PTCh. 11 - Prob. 12PTCh. 11 - Prob. 13PTCh. 11 - Prob. 14PTCh. 11 - Prob. 15PTCh. 11 - Prob. 16PTCh. 11 - Prob. 17PTCh. 11 - Prob. 18PTCh. 11 - Prob. 19PTCh. 11 - Prob. 20PTCh. 11 - Prob. 21PTCh. 11 - Prob. 22PTCh. 11 - Prob. 23PTCh. 11 - Prob. 24PTCh. 11 - Prob. 1ECh. 11 - Prob. 2ECh. 11 - Prob. 1STPCh. 11 - Prob. 2STPCh. 11 - Prob. 3STPCh. 11 - Prob. 4STPCh. 11 - Prob. 5STPCh. 11 - Prob. 6STPCh. 11 - Prob. 7STPCh. 11 - Prob. 8STPCh. 11 - Prob. 9STPCh. 11 - Prob. 10STPCh. 11 - Prob. 11STPCh. 11 - Prob. 12STPCh. 11 - Prob. 13STPCh. 11 - Prob. 14STPCh. 11 - Prob. 15STPCh. 11 - Prob. 16STPCh. 11 - Prob. 17STPCh. 11 - Prob. 18STP
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