   Chapter 9.7, Problem 55ES ### 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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# (For students who have studied calculus.) Explain how the equation below follows from the binomial theorem:    ( 1 + x ) n = ∑ k = 0 n ( n k ) x k . Write the formula obtained by taking thederivative of both sides of the equationin part (a) with respect to x. Use the result of part (b) to derivative the formulas below. (i)   2 n − 1 = 1 n [ ( n 1 ) + 2 ( n 2 ) + 3 ( n 3 ) + ⋅ ⋅ ⋅ + n ( n n ) ] (ii)   ∑ k = 0 n k ( n k ) ( − 1 ) k = 0 Express ∑ k = 1 n k ( n k ) 3 k in closed form (without using a summation sign or ellipsis).

To determine

(a)

To explain:

The given equation follows from binomial theorem.

Explanation

Given information:

(1+x)n=k=0n( n k)xk

Theorem used:

Binomial theorem: k=0n( n k)a(nk)bk=(a+b)n

Calculation:

Using binomial theorem with a=1,b=x:

To determine

(b)

To write:

The formula obtained by taking the derivative of both sides of the given equation with respect to x.

To determine

(c)

To derive:

(i).2n1=1n[( n 1 )+2( n 2 )+3( n 3 )+...+n( n n )](ii).k=1nk( n k )(1)k=0

To determine

(d)

To express:

k=1nk( n k)3k in closed form.

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