Explanation Approach: To prove the above expression for every integer n > 0 using mathematical induction, first show that the expression is valid for n = 0 . Then assume it to be true for n = k and at last show that the expression is valid for n = ( k + 1 ) . Calculation: Consider S n = a 0 + a 1 + a 3 + . . . + a n = 1 − a n + 1 1 − a First check the formula for n = 0 . S 0 = 1 = 1 − a 0 + 1 1 − a = 1 So the formula is valid for n = 0 . Assume that the formula is true for n = k . S k = a 0 + a 1 + a 3 + . . . + a k = 1 − a k + 1 1 − a Now show that the formula is valid for n = ( k + 1 ) . S k + 1 = 1 − a k + 2 1 − a Write S k + 1 as the sum of S k and ( k + 1 ) th term, a k + 1 . S k + 1 = a 0 + a 1 + a 3 +

To prove: a 0 + a 1 + a 3 + . . . + a n = 1 − a n + 1 1 − a , a ≠ 1 for every integer n > 0 , by using mathematical induction.

Elementary Linear Algebra

8th Edition

ISBN: 9780357156100

Author: Ron Larson

Publisher: Cengage Limited

See similar textbooks

Concept explainers

Permutations and Combinations

If there are 5 dishes, they can be relished in any order at a time. In permutation, it should be in a particular order. In combination, the order does not matter. Take 3 letters a, b, and c. The possible ways of pairing any two letters are ab, bc, ac, ba…

Counting Theory

The fundamental counting principle is a rule that is used to count the total number of possible outcomes in a given situation.

Videos

Question

Chapter A, Problem 9E

To determine

To prove:

a0+a1+a3+. . . +an=1−an+11−a ,a≠1 for every integer n>0, by using mathematical induction.

Expert Solution & Answer

Similar questions

Recommended textbooks for you

Elementary Linear Algebra (MindTap Course List)
Algebra
ISBN:9781305658004
Author:Ron Larson
Publisher:Cengage Learning
College Algebra
Algebra
ISBN:9781337282291
Author:Ron Larson
Publisher:Cengage Learning
College Algebra (MindTap Course List)
Algebra
ISBN:9781305652231
Author:R. David Gustafson, Jeff Hughes
Publisher:Cengage Learning
Elements Of Modern Algebra
Algebra
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Cengage Learning,
Algebra and Trigonometry (MindTap Course List)
Algebra
ISBN:9781305071742
Author:James Stewart, Lothar Redlin, Saleem Watson
Publisher:Cengage Learning
College Algebra
Algebra
ISBN:9781305115545
Author:James Stewart, Lothar Redlin, Saleem Watson
Publisher:Cengage Learning

Elementary Linear Algebra (MindTap Course List)

Algebra

ISBN:9781305658004

Author:Ron Larson

Publisher:Cengage Learning

College Algebra

Algebra

ISBN:9781337282291

Author:Ron Larson

Publisher:Cengage Learning

College Algebra (MindTap Course List)

Algebra

ISBN:9781305652231

Author:R. David Gustafson, Jeff Hughes

Publisher:Cengage Learning

Elements Of Modern Algebra

Algebra

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Cengage Learning,

Algebra and Trigonometry (MindTap Course List)

Algebra

ISBN:9781305071742

Author:James Stewart, Lothar Redlin, Saleem Watson

Publisher:Cengage Learning

College Algebra

Algebra

ISBN:9781305115545

Author:James Stewart, Lothar Redlin, Saleem Watson

Publisher:Cengage Learning

SEE MORE TEXTBOOKS

Find number of persons in a part with 66 handshakes Combinations; Author: Anil Kumar;https://www.youtube.com/watch?v=33TgLi-wp3E;License: Standard YouTube License, CC-BY

Discrete Math 6.3.1 Permutations and Combinations; Author: Kimberly Brehm;https://www.youtube.com/watch?v=J1m9sB5XZQc;License: Standard YouTube License, CC-BY

How to use permutations and combinations; Author: Mario's Math Tutoring;https://www.youtube.com/watch?v=NEGxh_D7yKU;License: Standard YouTube License, CC-BY

Permutations and Combinations | Counting | Don't Memorise; Author: Don't Memorise;https://www.youtube.com/watch?v=0NAASclUm4k;License: Standard Youtube License

Permutations and Combinations Tutorial; Author: The Organic Chemistry Tutor;https://www.youtube.com/watch?v=XJnIdRXUi7A;License: Standard YouTube License, CC-BY

To prove: a 0 + a 1 + a 3 + . . . + a n = 1 − a n + 1 1 − a , a ≠ 1 for every integer n > 0 , by using mathematical induction.

Solution Summary: The author explains how to prove the above expression for every integer n>0 using mathematical induction.

Concept explainers

Videos

To prove:

a0+a1+a3+. . . +an=1−an+11−a ,a≠1 for every integer n>0, by using mathematical induction.

Chapter A Solutions

To prove: a 0 + a 1 + a 3 + . . . + a n = 1 − a n + 1 1 − a , a ≠ 1 for every integer n &gt; 0 , by using mathematical induction.

Solution Summary: The author explains how to prove the above expression for every integer n>0 using mathematical induction.

Concept explainers

Videos

To prove: a0+a1+a3+. . . +an=1−an+11−a ,a≠1 for every integer n>0, by using mathematical induction.

Chapter A Solutions

To prove: a 0 + a 1 + a 3 + . . . + a n = 1 − a n + 1 1 − a , a ≠ 1 for every integer n > 0 , by using mathematical induction.

To prove:

a0+a1+a3+. . . +an=1−an+11−a ,a≠1 for every integer n>0, by using mathematical induction.