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Let x, y ≤ R and n € N. Use induction to prove the Binomial Theorem: (^.) x^yn-k. k n (x+y)² = Σ - k=0

Let x, y ≤ R and n € N. Use induction to prove the Binomial Theorem: (^.) x^yn-k. k n (x+y)² = Σ - k=0

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter2: The Integers
Section2.8: Introduction To Cryptography (optional)
Problem 23E
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Recall that for every non-negative integer n the factorial is defined as n! = 1.2... (n-1).n. In particular, 0! = 1 and n! =(n − 1)! · n for n ≥ 1. For every integer 0 ≤ k ≤n, we now define the binomial coefficient (2) as We moreover set (2) = 0 if k >n> 0. n k = n! k!(n − k)!* Let x, y E R and n € IN. Use induction to prove the Binomial Theorem: Jakyn-k. *** n (x + y)" =Σ( k=0 k REMARK. You are allowed to use basic rules to manipulate sums like the following without proof: if a₁,..., an, b₁,..., bn € R, then Σk=1 ak + Σk=1bk = Σk=1(ak + bk).
Transcribed Image Text:Recall that for every non-negative integer n the factorial is defined as n! = 1.2... (n-1).n. In particular, 0! = 1 and n! =(n − 1)! · n for n ≥ 1. For every integer 0 ≤ k ≤n, we now define the binomial coefficient (2) as We moreover set (2) = 0 if k >n> 0. n k = n! k!(n − k)!* Let x, y E R and n € IN. Use induction to prove the Binomial Theorem: Jakyn-k. *** n (x + y)" =Σ( k=0 k REMARK. You are allowed to use basic rules to manipulate sums like the following without proof: if a₁,..., an, b₁,..., bn € R, then Σk=1 ak + Σk=1bk = Σk=1(ak + bk).
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