   Chapter 2.2, Problem 17E

Chapter
Section
Textbook Problem

Use mathematical induction to prove that the stated property of the sigma notation is true for all positive integers n . (This sigma notation is defined in Section 1.6 .)a. a ∑ i = 1 n b i = ∑ i = 1 n a b i b. ∑ i = 1 n ( a i + b i ) = ∑ i = 1 n a i + ∑ i = 1 n b i

(a)

To determine

To prove: That ai=1nbi=i=1nabi is true for all n by using mathematical induction.

Explanation

Formula Used:

Considering the given statement is Pn for all integers n,

a. if Pn is true for n=1

b. if the truth of Pk always implies that Pk+1 is true, then the statement Pn is true for all positive integers n.

Proof:

For each positive integer n, let Pn be the statement

ai=1nbi=i=1nabi

When n=1, there is only one term, and no addition is actually performed.

When n=1, the value of the left side is

ai=1nbi=ab1

And the value of right side is

i=1nabi=ab1.

Thus P1 is true.

Assume now that Pk is true. That is, assume that the equation

ai=1kbi=i=1kabi is true.

With this assumption made, prove that Pk+1 is true. Add abk+1 to both sides of the assumed equality, the following result is obtained

(b)

To determine

To prove: That i=1n(ai+bi)=i=1nai+i=1nbi is true for all n by using mathematical induction.

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