The sum ∑ i = 1 n ( 2 i + 2 i ) .

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

Solutions

Chapter F, Problem 49E
To determine

Expert Solution

Answer to Problem 49E

The value of the sum i=1n(2i+2i) is 2n+1+n2+n2.

Explanation of Solution

The formula for the sum of a finite geometric serious with first term a and common ratio r1, is i=1nari1=a+ar+ar2++arn1=a(rn1)r1.

The sum i=1n(2i+2i) is simplified as i=1n2i+i=1n22i1.

Here, the sum i=1n22i1 is in the form i=1nari1 where, a=2andr=2.

By the above formula, the value of the sum i=1n22i1 is simplified as,

i=1n22i1=2[2n1]21=2(2n1)

Thus, the value of the sum i=1n22i1 is 2(2n1).

Simplify the expression i=1n(2i+2i) and obtain the value of the sum.

i=1n(2i+2i)=i=1n2i+i=1n22i1=2n(n+1)2+2(2n1)=n(n+1)+2(2n1)=2n+1+n2+n2

Thus, the value of the sum i=1n(2i+2i) is 2n+1+n2+n2.

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