(a) Prove that n n | k k k-1 | n > 2 and k < n. Hint: You do not need induction to prove this. Bear in mind that 0! = 1. I %3D (b) Verify that (") = 1 and (") by induction on n that () is an integer, for all k with 0 < k < n. (Note: You may have encountered (") as the count of the number of k element subsets of a set of n objects; it follows from this that (E) is an integer. What we are asking for here is an inductive proof based on algebra.) = 1. Use these facts, together with part (a), to prove %3D (c) Use part (a) and induction to prove the Binomial Theorem: For non-negative n and variables x, y, (2 + u)" = E (;)-*y*. %3D k k=0

(a) Prove that n n | k k k-1 | n > 2 and k < n. Hint: You do not need induction to prove this. Bear in mind that 0! = 1. I %3D (b) Verify that (") = 1 and (") by induction on n that () is an integer, for all k with 0 < k < n. (Note: You may have encountered (") as the count of the number of k element subsets of a set of n objects; it follows from this that (E) is an integer. What we are asking for here is an inductive proof based on algebra.) = 1. Use these facts, together with part (a), to prove %3D (c) Use part (a) and induction to prove the Binomial Theorem: For non-negative n and variables x, y, (2 + u)" = E (;)-y. %3D k k=0

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Description: (a) Prove thatnn|kkk-1|n > 2 and k < n. Hint: You do not need induction to prove this. Bear in mindthat 0! = 1.I%3D(b) Verify that (") = 1 and (")by induction on n that () is an integer, for all k with 0 < k < n. (Note: Youmay have encountered (") a

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter2: The Integers

Section2.2: Mathematical Induction

Problem 23E: Let and be integers, and let and be positive integers. Use mathematical induction to prove the...

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