   Chapter 9.7, Problem 14ES ### 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 that if n is an integer and n ≥ 1 , then 1 ⋅ 2 + 2 ⋅ 3 + ⋅ ⋅ ⋅ + n ( n + 1 ) = 2 ( 3 n + 2 ) .

To determine

To prove 12+23++n(n+1)=2( n+23), if n is an integer and n1.

Explanation

Given:

n is an integer and n1.

Concept used:

Pascal’s formula; ( n+1r)=(n r1)+(nr)

Proof:

Let’s show that the result is true for n=1 ;

Leftside=12=2

Right side=2( 1+23)=2(33)=2×3!3!0!=2

Therefore, for n=1, 12=2( 1+23).

Hence, the result is true for n=1.

Assume that the result is true for n=m. Therefore we can write;

12+23++m(m+1)=2( m+23)

Now, prove the proposition is true for n=m+1.

Add (m+1)(m+2) to both sides of 12+23++m(m+1)=2( m+23) and simplify the right side

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