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Give an algebraic proof (using known identities) of the identity m and n are nonnegative integers. m+ Σ (m + ^) - (m + n + 1). n where

Give an algebraic proof (using known identities) of the identity m and n are nonnegative integers. m+ Σ (m + ^) - (m + n + 1). n where

Advanced Engineering Mathematics
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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2. Give an algebraic proof (using known identities) of the identity m and n are nonnegative integers. Answer: 72 (**)-(m+n+¹) who where k
Transcribed Image Text:2. Give an algebraic proof (using known identities) of the identity m and n are nonnegative integers. Answer: 72 (**)-(m+n+¹) who where k
Expert Solution
Step 1

We prove this by induction on n.

Let P(n) : ∑k=0nm+kk=m+n+1n

Let n=0 , m+00=m!0!(m-0)!=1=(m+1)!0!(m+1-0)!=m+0+10

So P(0) is true.

Let n=1, ∑k=01m+kk=m0+m+11=1+(m+1)!(m+1-1)!=m+2=m+1+11

So P(1) is true.

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