n n+2 n n+r+1 ()+(†) + (7²) + + ( + ') - (^+²+¹) = 0 1 2 r r n
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.5: The Binomial Theorem
Problem 37E
Related questions
Question
Use induction to prove that
![Here is the transcription of the mathematical expression as seen in the image:
\[
\binom{n}{0} + \binom{n+1}{1} + \binom{n+2}{2} + \cdots + \binom{n+r}{r} = \binom{n+r+1}{r}
\]
This mathematical expression is a combinatorial identity used in binomial coefficient expansion and analysis. It denotes the sum of binomial coefficients starting from \(\binom{n}{0}\) up to \(\binom{n+r}{r}\), equating it to a single binomial coefficient \(\binom{n+r+1}{r}\).
**Explanation:**
This equation is essentially indicating that the sum of successive binomial coefficients is equal to another binomial coefficient. For example, in simple terms, it demonstrates that if we take a sum of binomial coefficients like so:
\[
\binom{n}{0} + \binom{n+1}{1} + \binom{n+2}{2} + \cdots + \binom{n+r}{r}
\]
It will always be equal to the binomial coefficient represented on the right-hand side of the equation:
\[
\binom{n+r+1}{r}
\]
This is a powerful identity in combinatorics and is useful in both theoretical and applied mathematics, especially in binomial expansions, probability, and various fields of discrete mathematics.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcea88763-6671-49a1-9b28-3ea76a79d4bc%2Fe74135ec-7e31-4720-baf8-3d852a219972%2Fhxxhsq_processed.png&w=3840&q=75)
Transcribed Image Text:Here is the transcription of the mathematical expression as seen in the image:
\[
\binom{n}{0} + \binom{n+1}{1} + \binom{n+2}{2} + \cdots + \binom{n+r}{r} = \binom{n+r+1}{r}
\]
This mathematical expression is a combinatorial identity used in binomial coefficient expansion and analysis. It denotes the sum of binomial coefficients starting from \(\binom{n}{0}\) up to \(\binom{n+r}{r}\), equating it to a single binomial coefficient \(\binom{n+r+1}{r}\).
**Explanation:**
This equation is essentially indicating that the sum of successive binomial coefficients is equal to another binomial coefficient. For example, in simple terms, it demonstrates that if we take a sum of binomial coefficients like so:
\[
\binom{n}{0} + \binom{n+1}{1} + \binom{n+2}{2} + \cdots + \binom{n+r}{r}
\]
It will always be equal to the binomial coefficient represented on the right-hand side of the equation:
\[
\binom{n+r+1}{r}
\]
This is a powerful identity in combinatorics and is useful in both theoretical and applied mathematics, especially in binomial expansions, probability, and various fields of discrete mathematics.
Expert Solution
![](/static/compass_v2/shared-icons/check-mark.png)
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 3 steps with 2 images
![Blurred answer](/static/compass_v2/solution-images/blurred-answer.jpg)
Recommended textbooks for you
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
![College Algebra](https://www.bartleby.com/isbn_cover_images/9781938168383/9781938168383_smallCoverImage.gif)
![College Algebra (MindTap Course List)](https://www.bartleby.com/isbn_cover_images/9781305652231/9781305652231_smallCoverImage.gif)
College Algebra (MindTap Course List)
Algebra
ISBN:
9781305652231
Author:
R. David Gustafson, Jeff Hughes
Publisher:
Cengage Learning
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
![College Algebra](https://www.bartleby.com/isbn_cover_images/9781938168383/9781938168383_smallCoverImage.gif)
![College Algebra (MindTap Course List)](https://www.bartleby.com/isbn_cover_images/9781305652231/9781305652231_smallCoverImage.gif)
College Algebra (MindTap Course List)
Algebra
ISBN:
9781305652231
Author:
R. David Gustafson, Jeff Hughes
Publisher:
Cengage Learning
![College Algebra](https://www.bartleby.com/isbn_cover_images/9781305115545/9781305115545_smallCoverImage.gif)
College Algebra
Algebra
ISBN:
9781305115545
Author:
James Stewart, Lothar Redlin, Saleem Watson
Publisher:
Cengage Learning
![Algebra and Trigonometry (MindTap Course List)](https://www.bartleby.com/isbn_cover_images/9781305071742/9781305071742_smallCoverImage.gif)
Algebra and Trigonometry (MindTap Course List)
Algebra
ISBN:
9781305071742
Author:
James Stewart, Lothar Redlin, Saleem Watson
Publisher:
Cengage Learning
![College Algebra](https://www.bartleby.com/isbn_cover_images/9781337282291/9781337282291_smallCoverImage.gif)