   Chapter 9.7, Problem 18ES ### 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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# Let m be any nonnegative integer. Use mathematical induction and Pascal’s formula to prove that for every integer n ≥ 0 , ( 0 m ) + ( 1 m + 1 ) + ... + ( n m + n ) = ( n m + n + 1 ) .

To determine

To prove using mathematical induction and Pascal’s formulathat for every integer n0 and a nonnegative integer m ,

(m0)+( m+11)++( m+nn)=( m+n+1n).

Explanation

Calculation:

(m0)+( m+11)++( m+nn)=( m+n+1n)

Prove that the result is true for n=0 by substitution.

Left side=(m0)=1.

Right side=( m+10)=1.

Therefore, (m0)=Left side=1=Right side=( m+10).

Hence the result is true for n=0.

Prove that the result is true for n=1 by substitution.

Left side=(m0)+( m+11)=( m+10)+( m+11)

Using Pascal’s law; Left side=( m+10)+( m+11)=( m+21)

Right side=( m+1+11)=( m+21)

Therefore, Left side=Right side=( m+21)

Hence the result is true for n=1

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