n n+2 n n+r+1 ()+(†) + (7²) + + ( + ') - (^+²+¹) = 0 1 2 r r n
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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.
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