Chapter 8 Sec 3 Geometric Sequences and Series

2 of 15 Pre Calculus Ch 8.3 Essential Question How do you find terms and sums of geometric sequences? Key Vocabulary: Geometric Sequence Common ratio

3 of 15 Pre Calculus Ch 8.3 Definition of a Geometric Sequence A sequence is geometric if the ratio of consecutive terms are the same. So, the sequence is geometric if there is a number r such that The number r is the common ratio of the sequence.

4 of 15 Pre Calculus Ch 8.3 Begin with n = 1. The common ratio is 2. The common ratio is 2. Example 1 a. The sequence whose n th term is 2 n is geometric. Begin with n = 1. The common ratio is 3. b. The sequence whose n th term is 4(3 n ) is geometric.. c. The sequence whose n th term is n 2 is NOT geometric.

5 of 15 Pre Calculus Ch 8.3 The n th term of a Geometric Sequence The n th term of an geometric sequence has the form Where r is the common ratio between consecutive terms of the sequence and every geometric sequence can be written in the following form. If you know the n th term, you can find the (n + 1) th term by multiplying by r.

6 of 15 Pre Calculus Ch 8.3 Example 2 Write the first five terms of the geometric sequence whose common ratio is r = 2 and whose first term is a 1 = 3. a 1 = 3 a 2 = 3(2 1 )=6 a 3 = 3(2 2 )=12 a 4 = 3(2 3 )=24 a 5 = 3(2 4 )=48

7 of 15 Pre Calculus Ch 8.3 Example 3 The 15 th term of the geometric sequence where a 1 = 20 and r = a n = a 1 r n – 1 a 15 = 20 (1.05) 15 – 1 a 15 = 20 (1.05) 14

8 of 15 Pre Calculus Ch 8.3 Example 4 Find a formula for the n th term of the following geometric sequence and what is the ninth term of the sequence? 5, 15, 45, …

9 of 15 Pre Calculus Ch 8.3 Example 5 The 4 th term of a geometric sequence is 125 and the 10 th term is 125/64. Find the 14 th term. (Assume that the terms are all positive.)

10 of 15 Pre Calculus Ch 8.3 The Sum of a Finite Geometric Sequence The Sum of a Finite Geometric Sequence with common ratio r ≠ 1 is given by Find the sum a 1 = 4(0.3), r = 0.3, and n = 12… When using this formula, be careful to check index begins at i = 1. If not then you’ll need to adjust sum.

11 of 15 Pre Calculus Ch 8.3 Geometric Series The Sum of a Infinite Geometric Series If | r | < 1, then the infinite geometric series has the sum Note: if | r | ≥ 1 the series has no sum. The sum of an infinite geometric sequence is called infinite geometric series or geometric series. If r has the property that | r | < 1, then it can be shown r n become arbitrarily close to zero as n increases without bound.

12 of 15 Pre Calculus Ch 8.3 Example 7 Find the sixth partial sum of the series, then find the sum of the series. a 1 = 4 and r = 0.6

13 of 15 Pre Calculus Ch 8.3 Example 8 Find the sum … a 1 = 3 and Compounded Interest Formulas Compounded Daily, Quarterly,… Compounded Continuously (only)

14 of 15 Pre Calculus Ch 8.3 Essential Question How do you find terms and sums of geometric sequences?

15 of 15 Pre Calculus Ch 8.3 Daily Assignment Chapter 8 Section 3 Text Book Pg 607 – 608 All ODD OF: #1 – 21; #33 – 35; #55 – 65; #73 – 75; #83 – 85 Read Section 8.5 Show all work for credit.