Evaluate (1) − ² (17) + ³ (2) + - 2 ++ (-1)(n+1) 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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![### Problem 16: Evaluate the Following Expression
Evaluate the summation:
\[
\binom{n}{0} - 2\binom{n}{1} + 3\binom{n}{2} - \cdots + (-1)^n (n+1) \binom{n}{n}
\]
**Explanation**:
- The expression involves binomial coefficients, denoted by \(\binom{n}{k}\), which represent the number of ways to choose \(k\) items from a set of \(n\) items without regard to order.
- The terms are alternately added and subtracted, following a pattern where the \(k\)-th term is multiplied by \((-1)^k(k+1)\).
- This creates an alternating sum of binomial coefficients multiplied by consecutive integers.
Understanding this problem and finding the evaluation requires knowledge of binomial theorem properties and summation techniques.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7011b814-0888-4e7c-ad26-86019bf83fb9%2F8e2605ca-5b01-47ba-bc84-b9bb98b888ba%2Fh8bhge_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem 16: Evaluate the Following Expression
Evaluate the summation:
\[
\binom{n}{0} - 2\binom{n}{1} + 3\binom{n}{2} - \cdots + (-1)^n (n+1) \binom{n}{n}
\]
**Explanation**:
- The expression involves binomial coefficients, denoted by \(\binom{n}{k}\), which represent the number of ways to choose \(k\) items from a set of \(n\) items without regard to order.
- The terms are alternately added and subtracted, following a pattern where the \(k\)-th term is multiplied by \((-1)^k(k+1)\).
- This creates an alternating sum of binomial coefficients multiplied by consecutive integers.
Understanding this problem and finding the evaluation requires knowledge of binomial theorem properties and summation techniques.
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