Geometric Sequences and Series Section 11.3 Geometric Sequences and Series Copyright ©2013, 2009, 2006, 2005 Pearson Education, Inc.

Objectives Identify the common ratio of a geometric sequence, and find a given term and the sum of the first n terms. Find the sum of an infinite geometric series, if it exists.

Geometric Sequences A sequence in which each term after the first is found by multiplying the preceding term by the same number is a geometric sequence. The number that is multiplied by each term to produce the next term is called the common ratio. A sequence is geometric if there is a number r, called the common ratio, such that or an+1 = anr, for any integer n  1.

Example For each of the following geometric sequences, identify the common ratio. 4, 16, 64, 256, 1024, … , … $750, $600, $480, $384, … Solution: To find the common ratio, we divide any term (other than the first) by the preceding term.

Example continued 4, 16, 64, 256, 1024, … b) , … 0.8 ( , and so on) $750, $600, $480, $384, … , and so on) b) , … 4 ( and so on) 4, 16, 64, 256, 1024, … Common Ratio Sequence

nth Term of a Geometric Sequence The nth term of a geometric sequence is given by the formula: an = a1r n  1, for any integer

Example Find the 11th term of the geometric sequence 1, 3, 9, 27, … Solution: We first note that a1 = 1, and n = 11. We then find the common ratio or 3. Then using the formula an = a1r n  1, we have The 11th term is 59,049.

Sum of the First n Terms The formula for the sum of the first n terms of a geometric sequence is given by: , for any r  1.

Example Find the sum: This is a geometric series with a1 = 6, r = 3, and n = 12.

Infinite Geometric Series The sum of the terms of an infinite geometric sequence is an infinite geometric series. For some geometric sequences, Sn gets close to a specific number as n gets large. Consider the infinite series + … . Each of the partial sums is less than 1, but Sn gets very close to 1 as n gets large.

Infinite Geometric Series continued We say that 1 is the limit of Sn and also that 1 is the sum of the infinite geometric sequence. The sum of an infinite geometric sequence is denoted S . In this case S = 1.

Limit or Sum of an Infinite Geometric Series When , the limit or sum of an infinite geometric series is given by

Example Determine whether each of the following infinite geometric series has a limit. If a limit exists, find it. a) 3 + 6 + 12 + 24 +    b)

Example (cont) a) 3 + 6 + 12 + 24 +    Here r = 2, so Since , the series does not have a limit. Here , so . Since , the series does have a limit. We find the limit:

Example A racquetball hit in the air 27 feet rebounds to two-thirds of its previous height after each bounce. Find the total vertical distance the ball has traveled when it hits the ground the tenth time. Solution: Between bounces, the racquetball travels twice its rebound height. The geometric series for the total distance traveled is 54 + 36 + 24 +    .

Example continued Using a1 = 54 and r = The ball has traveled about 159.2 feet by the tenth bounce.