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11.3 Geometric Sequences and Series. A geometric sequence (or geometric progression ) is a sequence in which each term after the first is obtained by multiplying the preceding term by a constant nonzero real number, called the common ratio .
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11.3 Geometric Sequences and Series • A geometric sequence (or geometric progression) is a sequence in which each term after the first is obtained by multiplying the preceding term by a constant nonzero real number, called the common ratio. • The series of wages 1, 2, 4, 8, 16 … is an example of a geometric sequence in which the first term is 1 and the common ratio is 2.
11.3 Finding the Common Ratio • In a geometric sequence, the common ratio can be found by choosing any term except the first and dividing by the preceding term. The geometric sequence 2, 8, 32, 128, … has common ratio r = 4 since
11.3 Geometric Sequences and Series nth Term of a Geometric Sequence In a geometric sequence with first term a1 and common ratio r, neither of which is zero, the nth term is given by
11.3 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
Your turn. Find a5 and an. 1. 2. 3.
11.3 Modeling a Population of Fruit Flies Example A population of fruit flies is growing in such a way that each generation is 1.5 times as large as the last generation. Suppose there were 100 insects in the first generation. How many would there be in the fourth generation? Solution The populations form a geometric sequence with a1= 100 and r = 1.5 . Use n = 4 in the formula for an.. In the fourth generation, the population is about 338 insects.
11.3 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
11.3 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:
11.3 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 , .
Practice Find the sum of the finite geometric series. 1. 2. 3.
11.3 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
11.3 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 .
11.3 Finding Sums of the Terms of Infinite Geometric Sequences Example Find Solution Here and so .
Infinite Geometric Series Practice Find the sum of the series. 1. 2.
11.3 Annuities Future Value of an Annuity The formula for the future value of an annuity is where S is the future value, R is the payment at the end of each period, i is the interest rate in decimal form per period, and n is the number of periods.
The Value of an Annuity You have an annuity with a monthly payment of $250 that pays a annual interest rate of 6%. How much will it be worth after 10 years? i = .5% = .005 n = 120
