   Chapter 9.7, Problem 15ES ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193

#### Solutions

Chapter
Section ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193
Textbook Problem
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# Prove the following generalization of exercise 13: Let r be a fixed nonnegative integer. For every integer n with n ≥ r , ∑ t = 2 n ( i i ) = ( r + 1 n + 1 )

To determine

To prove for every integer n with nr, i=rn( i r)=(n+1r+1).

Explanation

Given information:

Let r be a fixed nonnegative integer and nr.

Proof:

The formula can be proved using the method of mathematical induction.

Let P(n) be “ i=rn( i r)=(n+1r+1) ”.

Basis stepn=r

i=rn( i r )=( r r)=r!r!( rr)!=r!r!0!=1

Also, ( n+1 r+1)=( r+1 r+1)=( r+1)!( r+1)!( ( r+1 )( r+1 ))!=( r+1)!( r+1)!0!=1

Thus P(r) is true.

Inductive step Let P(k) be true, thus i=rk( i r)=(n+1r+1) with kr

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