11-5 Geometric Series Hubarth Algebra II

A geometric series is the expression for the sum of the terms of a geometric sequence. As with arithmetic series, you can use

The first term is 5, and there are six terms in the series. The sum of the series is The common ratio is = = = = = So a 1 = 5, r = 3, and n = 6. S n = Write the formula. a 1 (1 – r n ) 1 – r = Substitute a 1 = 5, r = 3, and n = 6. 5 (1 – 3 6 ) 1 – 3 = = 1820 Simplify. –3640 –2 Ex. 1 Using the Geometric Series Formula Use the formula to evaluate the series

Ex. 2 Real-World Connection The Floyd family starts saving for a vacation that is one year away. They start with $125. Each month they save 8% more than the previous month. How much money will they have saved 12 months later? Relate: S n = a 1 (1 – r n ) 1 – r Write the formula for the sum of a geometric series. The amount of money the Floyd’s will have saved will be $ Define: S n = total amount saved a 1 = 125Initial amount. r = 1.08Common ratio. n = 12Number of months. Write: S 12 = 125 ( 1 – ) 1 – Use a calculator. Substitute.

Ex. 3 Determining Divergence and Convergence Decide whether each infinite geometric series diverges or converges. Then determine whether the series has a sum.   n = n a. b a 1 = =, a 2 = = a 1 = 2, a 2 = 6 r = ÷ = r = 6 ÷ 2 = 3 Since | r | < 1, the series converges, and the series has a sum. Since | r | 1, the series diverges, and the series does not have a sum. >

The weight at the end of a pendulum swings through an arc of 30 inches on its first swing. After that, each successive swing is 85% of the length of the previous swing. What is the total distance the weight will swing by the time it comes to rest? The largest arc the pendulum swings through is on the first swing of 30 in., so a 1 = 30. = 200Simplify. The total distance that the pendulum swings through is 200 in. S = Use the formula. a 1 1 – r = Substitute – 0.85 Ex. 4 Real-World Connection

Practice converge diverges 2 2