Ch 9.3 Geometric Sequences and Series Lessons

Transcription

1 Ch 9.3 Geometric Sequeces ad Series Lessos SKILLS OBJECTIVES Recogize a geometric sequece. Fid the geeral, th term of a geometric sequece. Evaluate a fiite geometric series. Evaluate a ifiite geometric series, if it exists. Use geometric sequeces ad series to model real-world problems. CONCEPTUAL OBJECTIVES Uderstad the differece betwee a geometric sequece ad a geometric series. Distiguish betwee a arithmetic sequece ad a geometric sequece. Uderstad why it is ot possible to evaluate all ifiite geometric series. Geometric Sequeces I Sectio 9.2, we discussed arithmetic sequeces, where successive terms had a commo differece. I other words, each term was foud by addig the same costat to the previous term. I this sectio we discuss geometric sequeces, where successive terms have a commo ratio. I other words, each term is foud by multiplyig the previous term by the same costat. The sequece 4, 12, 36, is geometric because each successive term is foud by multiplyig the previous term by 3.

2 DEFINITION Geometric Sequeces A sequece is geometric if each term i the sequece is foud by multiplyig the previous term by a umber r, so that 1 a 1 r a = + r a. Because = a+, the umber r is called the commo ratio. EXAMPLE 1 Idetifyig the Commo Ratio i Geometric Sequeces Fid the commo ratio for each of the geometric sequeces. a. 5, 20, 80, 320,...

3 Fid the commo ratio for each of the geometric sequeces b. 1,,,,

4 c. $5.000, $5,500, $6,050, $6, 655,... YOUR TURN Fid the commo ratio of each geometric series. a. 1, a. solutio: -3 b. 320, 80, 20, 5,... b. solutio 1 or

5 The Geeral (th) Term of a Geometric Sequece To fid a formula for the geeral, or th, term of a geometric sequece, write out the first several terms ad look for a patter. THE TH TERM OF A GEOMETRIC SEQUENCE The th term of a geometric sequece with commo ratio r is give by 1 a = a r or by 1 for 1 a = 1 a + 1 r for 0

6 EXAMPLE 2 Fidig the th Term of a Geometric Sequece Fid the 7th term of the sequece 2, 10, 50, 250

7 YOUR TURN Fid the 8th term of the sequece 3, 12, 48, 192 Aswer: 49,152 EXAMPLE 3 Fidig the Geometric Sequece Fid the geometric sequece whose 5th term is 0.01 ad whose commo ratio is 0.1. YOUR TURN Fid the geometric sequece whose 4th term is 3 ad whose commo ratio is 1 3. Aswer: 81, 27, 9, 3, 1...

8 Geometric Series The sum of the terms of a geometric sequece is called a geometric series. If we oly sum the first terms of a geometric sequece, the result is a fiite geometric series give by To develop a formula for the th partial sum, we multiply the above equatio by r: Subtractig the secod equatio from the first equatio, we fid that all of the terms o the right side drop out except the first term i the first equatio ad the last term i the secod equatio: Cotiued o ext page.

9 Factor the S, out of the left side ad the a 1, out of the right side: Divide both sides by (1- r), assumig r 1. The result is a geeral formula for the sum of a fiite geometric series: 1 ( 1 r ) S = a r 1 k = 1 ( 1 r) S = a r = a + a r+ a r + a r + + a r k Study Tip The uderscript k = 1 applies oly whe the summatio starts at the a 1 term. It is importat to ote which term is the startig term.

10 EXAMPLE 4 Evaluatig a Fiite Geometric Series Evaluate the fiite geometric series.

11 b. The first ie terms of the series

12 The sum of a ifiite geometric sequece is called a ifiite geometric series. Some ifiite geometric series coverge (yield a fiite sum) ad some diverge (do ot have a fiite sum). For example, For ifiite geometric series that coverge, the partial sum S, approaches a sigle umber as gets large. The formula used to evaluate a fiite geometric series ca be exteded to a ifiite geometric series for certai values of,: If r < 1, the whe r is raised to a power, it cotiues to get smaller, approachig 0. For those values of r, ifiite geometric series coverges to a fiite sum.

13 EVALUATING AN INFINITE GEOMETRIC SERIES The sum of a ifiite geometric series is give by the formula 1 a1 r = a1 r < 1 1 r = 0 ( ) Study Tip The formula used to evaluate a ifiite geometric series is: First term 1 Ratio EXAMPLE 5 Determiig Whether the Sum of a Ifiite Series Exists Determie whether the sum exists for each of the geometric series. a

14 Determie whether the sum exists for each of the geometric series. YOUR TURN Determie whether the sum exists for the followig geometric series. Aswer: a. yes b. o Do you expect ad + + to sum to the same umber? The aswer is o, because the secod series is a alteratig series ad terms are both added ad subtracted. Hece, we would expect the secod series to sum to a smaller umber tha the first series sums to.

15 EXAMPLE 6 Evaluatig a Ifiite Geometric Series Evaluate each ifiite geometric series.

16 Notice that the alteratig series summed to 3 4, whereas the positive series summed to 3 2. YOUR TURN Fid the sum of each ifiite geometric series.

17 It is importat to ote the restrictio o the commo ratio r. The absolute value of the commo ratio has to be strictly less tha 1 for a ifiite geometric series to coverge. Otherwise the ifiite geometric series diverges.

18 EXAMPLE 7 Evaluatig a Ifiite Geometric Series Evaluate the ifiite geometric series, if possible.

19 Applicatios Suppose you are give a job offer with a guarateed percetage raise per year. What will your aual salary be 10 years from ow? That aswer ca be obtaied usig a geometric sequece. Suppose you wat to make volutary cotributios to a retiremet accout directly debited from your paycheck every moth. Suppose the accout ears a fixed percetage rate: 1. How much will you have i 30 years if you deposit $50 a moth? 2. What is the differece i the total you will have i 30 years if you deposit $100 a moth istead? These importat questios about your persoal fiaces ca be aswered usig geometric sequeces ad series. EXAMPLE B Future Salary: Geometric Sequece Suppose you are offered a job as a evet plaer for the PGA Tour. The startig salary is $45,000, ad employees are give a 5% raise per year. What will your aual salary be durig the 10th year with the PGA Tour?

20 YOUR TURN Suppose you are offered a job with AT&T at $37,000 per year with a guarateed raise of 4% after every year. What will your aual salary be after 15 years with the compay? Aswer: $64,072.03

21 EXAMPLE 9 Savigs Growth: Geometric Series Kare has maitaied acrylic ails by payig for them with moey eared from a part-time job. After hearig a lecture from her ecoomics professor o the importace of ivestig early i life, she decides to remove the acrylic ails, which cost $50 per moth, ad do her ow maicures. She has that S50 automatically debited from her checkig accout o the first of every moth ad put ito a moey market accout that receives 3% iterest compouded mothly. What will the balace be i the moey market accout exactly 2 years from the day of her iitial $50 deposit?

22 YOUR TURN Repeat Example 9 with Kare puttig $100 (istead of $50) i the same moey market. Assume she does this for 4 years (istead of 2 years). Aswer: $

23 Sectio 9.3 Summary I this sectio, we discussed geometric sequeces, i which each successive term is foud by multiplyig the previous term by a costat, so that a = 1 r + a. That costat, r, is called the commo ratio. The th term of a geometric sequece is give by 1 a = a r or by 1 for 1 a = 1 a + 1 r for 0 The sum of the terms of a geometric sequece is called a geometric series. Fiite geometric series coverge to a umber. Ifiite geometric series coverge to a umber if the absolute value of the commo ratio is less tha 1. If the absolute value of the commo ratio is greater tha or equal to 1, the ifiite geometric series diverges ad the sum does ot exist. May real-world applicatios ivolve geometric sequeces ad series, such as growth of salaries ad auities through percetage icreases.

24 Sectio 9.3 Summary cotiued Fiite geometric series: 1 ( 1 r ) S = a r 1 k = 1 ( 1 r) S = a r = a + a r+ a r + a r + + a r k EVALUATING AN INFINITE GEOMETRIC SERIES The sum of a ifiite geometric series is give by the formula 1 a1 r = a1 r < 1 1 r = 0 ( ) Next sectio is 9.3 Exercises