(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
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Topic Video
Question
![(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](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbad686e8-54b0-4143-baf2-8fe7ab009e74%2Fe5ddb44d-82e6-47ad-9e46-bd3faaa34b78%2Fqtj170t.png&w=3840&q=75)
Transcribed Image Text:(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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