9.3 Geometric Sequences and Series A geometric sequence is a sequence in which each term after the first is obtained by multiplying the preceding term by a constant nonzero real number. 1, 2, 4, 8, 16 … is an example of a geometric sequence with first term 1 and each subsequent term is 2 times the term preceding it. The multiplier from each term to the next is called the common ratio and is usually denoted by r.

9.3 Finding the Common Ratio In a geometric sequence, the common ratio can be found by dividing any term by the term preceding it. The geometric sequence 2, 8, 32, 128, … has common ratio r = 4 since

9.3 Geometric Sequences and Series nth Term of a Geometric Sequence In the geometric sequence with first term a1 and common ratio r, the nth term an, is

9.3 Using the Formula for the nth Term Example Find a5 and an for the geometric sequence 4, –12, 36, –108 , … Solution Here a1= 4 and r = 36/ –12 = – 3. Using n=5 in the formula In general

9.3 Modeling a Population of Fruit Flies Example A population of fruit flies grows in such a way that each generation is 1.5 times the previous generation. There were 100 insects in the first generation. How many are in the fourth generation. Solution The populations form a geometric sequence with a1= 100 and r = 1.5 . Using n=4 in the formula for an gives or about 338 insects in the fourth generation.

9.3 Geometric Series A geometric series is the sum of the terms of a geometric sequence . In the fruit fly population model with a1 = 100 and r = 1.5, the total population after four generations is a geometric series:

9.3 Geometric Sequences and Series Sum of the First n Terms of an Geometric Sequence If a geometric sequence has first term a1 and common ratio r, then the sum of the first n terms is given by where .

9.3 Finding the Sum of the First n Terms Example Find Solution This is the sum of the first six terms of a geometric series with and r = 3. From the formula for Sn , .

9.3 Infinite Geometric Series If a1, a2, a3, … is a geometric sequence and the sequence of sums S1, S2, S3, …is a convergent sequence, converging to a number S. Then S is said to be the sum of the infinite geometric series

9.3 An Infinite Geometric Series Given the infinite geometric sequence the sequence of sums is S1 = 2, S2 = 3, S3 = 3.5, … The calculator screen shows more sums, approaching a value of 4. So

9.3 Infinite Geometric Series Sum of the Terms of an Infinite Geometric Sequence The sum of the terms of an infinite geometric sequence with first term a1 and common ratio r, where –1 < r < 1 is given by .

9.3 Finding Sums of the Terms of Infinite Geometric Sequences Example Find Solution Here and so .