To prove: The formula for the sum of a finite geometric serious with first term a and common ratio r ≠ 1 , ∑ i = 1 n a r i − 1 = a + a r + a r 2 + ⋯ + a r n − 1 = a ( r n − 1 ) r − 1 .

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 47E
To determine

To prove: The formula for the sum of a finite geometric serious with first term a and common ratio r≠1, ∑i=1nari−1=a+ar+ar2+⋯+arn−1=a(rn−1)r−1.

Expert Solution

Explanation of Solution

Definition used:

If am,am+1,...,an are real numbers and m and n are integers such that mn, then i=mnai=am+am+1+am+2++an1+an.

Calculation:

By the above definition, the expression i=1nari1 simplified as follows.

i=1nari1=ar(11)+ar(21)+ar(31)++ar(n1)=a+ar1+ar2++arn1

That is, i=1nari1=a+ar1+ar2++arn1 (1)

Consider S=a+ar1+ar2++arn1 (2)

Multiply both side of the equation (2) by r gives,

rS=r(a+ar1+ar2++arn1)=ar+ar2+ar3++arn

That is, rS=ar+ar2+ar3++arn (3)

Subtract equation (2) from the equation (3).