1. Chapter 11 Sequences, Induction, and Probability Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1 11.3 Geometric Sequences and Series

2. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 2 Find the common ratio of a geometric sequence. Write terms of a geometric sequence. Use the formula for the general term of a geometric sequence. Use the formula for the sum of the first n terms of a geometric sequence. Find the value of an annuity. Use the formula for the sum of an infinite geometric series. Objectives:

3. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 3 Definition of a Geometric Sequence A geometric sequence is a sequence in which each term after the first is obtained by multiplying the preceding term by a fixed nonzero constant. The amount by which we multiply each time is called the common ratio of the sequence.

4. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 4 Example: Writing the Terms of a Geometric Sequence Write the first six terms of the geometric sequence with first term 12 and common ratio First term = 12 Second term Third term The terms are Fourth term Fifth term Sixth term

5. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 5 General Term of a Geometric Sequence The nth term (the general term) of a geometric sequence with first term a 1 and common ratio r is

6. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 6 Example: Using the Formula for the General Term of a Geometric Sequence Find the seventh term of the geometric sequence whose first term is 5 and whose common ratio is –3. n = 7, a 1 = 5, r = –3 The seventh term of the sequence is 3645.

7. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 7 Example: Writing the General Term for a Geometric Sequence Write the general term for the geometric sequence 3, 6, 12, 24, 48,... Then use the formula to find the eighth term. a 1 = 3 At the eighth term, n = 8. The eighth term is 384. The general term of the sequence is

8. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 8 The Sum of the First n Terms of a Geometric Sequence The sum, S n, of the first n terms of a geometric sequence is given by in which a 1 is the first term and r is the common ratio (r 1).

9. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 9 Example: Finding the Sum of the First n Terms of a Geometric Sequence Find the sum of the first nine terms of the geometric sequence:2, –6, 18, –54,... a 1 = 2 The sum is 9842.

10. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 10 Value of an Annuity: Interest Compounded n Times per Year If P is the deposit made at the end of each compounding period for an annuity at r percent annual interest compounded n times per year, the value, A, of the annuity after t years is

11. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 11 Example: Determining the Value of an Annuity At age 30, to save for retirement, you decide to deposit $100 at the end of each month into an IRA that pays 9.5% compounded monthly. How much will you have from the IRA when you retire at age 65? r = 0.095n = 12t = 65 – 30 = 35

12. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 12 Example: Determining the Value of an Annuity (continued) At age 30, to save for retirement, you decide to deposit $100 at the end of each month into an IRA that pays 9.5% compounded monthly. How much will you have from the IRA when you retire at age 65? At age 65, you will have $333,946. Find the interest. Interest = Value of IRA – Total deposits The interest is $291,946.

13. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 13 The Sum of an Infinite Geometric Series

14. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 14 Example: Finding the Sum of an Infinite Geometric Series Find the sum of the infinite geometric series: The sum of the series is 9.