(a) Prove that () = 1, (n) = 1, and (^)= (₁ = ¹) + n if 1≤k ≤n-1. (b) Use part (a) and induction to prove that (2) is a positive integer for all n, k = Z such that 0≤ k ≤n. (c) Let x, y E R. Prove that for every integer n ≥ 0, n (x + y)² = Σ (1) xn-kyk, k=0
(a) Prove that () = 1, (n) = 1, and (^)= (₁ = ¹) + n if 1≤k ≤n-1. (b) Use part (a) and induction to prove that (2) is a positive integer for all n, k = Z such that 0≤ k ≤n. (c) Let x, y E R. Prove that for every integer n ≥ 0, n (x + y)² = Σ (1) xn-kyk, k=0
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 5:**
For \( n, k \in \mathbb{Z} \) such that \( 0 \leq k \leq n \), define
\[
\binom{n}{k} = \frac{n!}{k!(n-k)!}.
\]
Note that, by convention, \( 0! = 1 \).
**(a)** Prove that \( \binom{n}{0} = 1 \), \( \binom{n}{n} = 1 \), and
\[
\binom{n}{k} = \binom{n-1}{k} + \binom{n-1}{k-1}
\]
if \( 1 \leq k \leq n-1 \).
**(b)** Use part (a) and induction to prove that \( \binom{n}{k} \) is a positive integer for all \( n, k \in \mathbb{Z} \) such that \( 0 \leq k \leq n \).
**(c)** Let \( x, y \in \mathbb{R} \). Prove that for every integer \( n \geq 0 \),
\[
(x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k}y^k.
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd7626d59-6405-4948-a625-19aca32c9eb7%2F8818dc3a-67f7-408c-bfde-267cced7a408%2Fun0dfqr_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 5:**
For \( n, k \in \mathbb{Z} \) such that \( 0 \leq k \leq n \), define
\[
\binom{n}{k} = \frac{n!}{k!(n-k)!}.
\]
Note that, by convention, \( 0! = 1 \).
**(a)** Prove that \( \binom{n}{0} = 1 \), \( \binom{n}{n} = 1 \), and
\[
\binom{n}{k} = \binom{n-1}{k} + \binom{n-1}{k-1}
\]
if \( 1 \leq k \leq n-1 \).
**(b)** Use part (a) and induction to prove that \( \binom{n}{k} \) is a positive integer for all \( n, k \in \mathbb{Z} \) such that \( 0 \leq k \leq n \).
**(c)** Let \( x, y \in \mathbb{R} \). Prove that for every integer \( n \geq 0 \),
\[
(x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k}y^k.
\]
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