12-2: Geometric Sequences and Series

1 Geometric Sequences and SeriesACCOUNTINGB ertha Blackwell is an accountant for a smallcompany. On January 1, 1996, the company purchased $50,000worth of office copiers. Since this equipment is a company asset, Ms. Blackwell needs to determine how much the copiers are presently worth. Sheestimates that copiers depreciate at a rate of 45% per year. What value should Ms. Blackwell assign the copiers on her 2001 year-end accounting report? Thisproblem will be solved in Example following sequence is an example of a Geometric , 2, , , , ..The ratio of successive terms in a Geometric sequence is a constant calledthe common ratio,denoted can find the next term in a Geometric sequence as follows.

2 First divide any term by the preceding term to find the common ratio. Then multiply the last term by the common ratio to find the next term inthe the common ratio and find the next three terms in 1, 12 , 14 , ..First, find the common a1 12 1 or 12 a3 a2 14 12 or 12 The common ratio is 12 .Multiply the third term by 12 to get the fourth term, and so 14 12 or 18 a5 18 12 or 116 a6 116 12 or 312 The next three terms are 18 , 116 , and 312 .766 Chapter 12 Sequences and Series12-2 RealWorldApplicationOBJECTIVES Find the nthterm andgeometricmeans of ageometricsequence. Find the sum ofnterms of ageometric Geometric sequence is a sequence in which each term after the first, a1, is the product of the preceding term and the common ratio, r.

3 The terms of the sequence can be represented as follows, where a1isnonzero and ris not equal to 1 or , a1r, a1r2, ..GeometricSequenceExample1 Can you find the next term? 1, 3r 3, 9r 9, ..First, find the common a1 r3 r 13 a3 a2 93rr 93 a2 a1 3r( r 11) a2 93(r(r 11)) a1 a2 common ratio is the third term by 3 to get the fourth term, and so 3(9r 9) or 27r 27a5 3( 27r 27 ) or 81r 81a6 3(81r 81) or 243r 243 The next three terms are 27r 27, 81r 81, and 243r with arithmetic Sequences , Geometric Sequences can also be definedrecursively. By definition, the nth term is also equal to an 1r, where an 1is the(n 1)th term. That is, an an successive terms of a Geometric sequence can be expressed as theproduct of the common ratio and the previous term, it follows that each term canbe expressed as the product of a1and a power of r.

4 The terms of a geometricsequence for which al 5 and r 7 can be represented as terma1a1 5second terma2a1r 5 71 35third terma3a1r2 5 72 245fourth terma4a1r3 5 73 1715fifth terma5a1r4 5 74 12,005 nth termanarn 1 5 7n 1 Find an approximation for the 23rd term in the sequence 256, , , ..First, find the common a1 256 or a2 ( ) or common ratio is , use the formula for the nth term of a Geometric a1r n 1a23 256( )23 1n 23, a1 256, r a 23rd term is about 12-2 Geometric Sequences and Series767 The nth term of a Geometric sequence with first term a1and commonratio ris given by an a1rn nth Term of aGeometricSequenceExample2 Geometric Sequences can represent growth or decay.

5 For a common ratio greater than 1, a sequence may model include compound interest, appreciation of property, and population growth. For a positive common ratio less than 1, a sequence may model include some radioactive behavior and to the application at thebeginning of the lesson. Compute the value of the copiers at the end of the year the copiers were purchased at thebeginning of the first year, the original purchaseprice of the copiers represents a1. If the copiersdepreciate at a rate of 45% per year, then theyretain 100 45 or 55% of their value each the formula for the nth term of a geometricsequence to find the value of the copiers sixyears later or a1rn 1a7 50,000 ( )7 1a1 50,000, r , n 7a7 a Blackwell should list the value of the copiers on her report as $ terms between any two nonconsecutive terms of a Geometric sequenceare called Geometric a sequence that has two Geometric means between 48 and sequence will have the form 48, ?

6 , ?, , find the common a1rn 1a4 a1r3 Since there will be four terms in the sequence , n 4. 750 48r3a4 750 and a1 48 1825 r3 Divide each side by 48 and simplify. 3 1285 rTake the cube root of each side. rThen, determine the Geometric 48( ) or 120a3 120( ) or 300 The sequence is 48, 120, 300, 12 Sequences and SeriesRealWorldApplicationExample3 Example4 The formula forthe sum of a Geometric seriescan also be written as Sn a11 rn 1 Geometric seriesis the indicated sum of the terms of a geometricsequence. The lists below show some examples of Geometric Sequences and their corresponding SequenceGeometric Series3, 9, 27, 81, 2433 9 27 81 24316, 4, 1, 14 , 116 16 4 1 14 116 a1, a2, a3, a4.

7 , ana1 a2 a3 a4 .. anTo develop a formula for the sum of a finite Geometric sequence , Sn, write anexpression for Snand for rSn, as shown below. Then subtract rSnfrom Snandsolve for a1 a1r a1r2 .. a1rn 2 a1rn 1rSn a1r a1r2 .. a1rn 2 a1rn 1 a1rnSn rSn a1 (1 r) a1 a11 ar1rn Divide each side by 1 r, r the sum of the first ten terms of the Geometric Series 16 48 144 432 ..First, find the common a1 48 16 or 3a4 a3 432 144 or 3 The common ratio is a11 ar1rn S10 n 10, a1 16, r 236,192 Use a sum of the first ten terms of the Series is 236, and other financial institutions use compound interest to determineearnings in accounts or how much to charge for loans.

8 The formula for compound interest is A P 1 nr tn, whereA the account balance,P the initial deposit or amount of money borrowed,r the annual percentage rate (APR),n the number of compounding periods per year, andt the time in 16( 3)10 1 ( 3)Lesson 12-2 Geometric Sequences and Series769 The sum of the first nterms of a finite Geometric Series is given by Sn a11 ar1rn .Sum of a FiniteGeometricSeriesExample5 Suppose at the beginning of each quarter you deposit $25 in a savingsaccount that pays an APR of 2% compounded quarterly. Most banks post theinterest for each quarter on the last day of the quarter. The chart below lists theadditions to the account balance as a result of each successive deposit through the rest of the year.

9 Note that 1 nr 1 or chart shows that the first deposit will gain interest through all fourcompounding periods while the second will earn interest through only threecompounding periods. The third and last deposits will earn interest through twoand one compounding periods, respectively. The sum of these amounts, $ ,is the balance of the account at the end of one year. This sum also represents afinite Geometric Series where a1 , r , and n ( ) ( )2 ( )3 INVESTMENTSH iroshi wants to begin saving money for college. He decidesto deposit $500 at the beginning of each quarter (January 1, April 1, July 1,and October 1) in a savings account that pays an APR of 6% compoundedquarterly.

10 The interest for each quarter is posted on the last day of thequarter. Determine Hiroshi s account balance at the end of one interest is compounded each quarter. So n 4 and the interest rate perperiod is 6% 4 or The common ratio rfor the Geometric Series is then 1 , or first term a1in this Series is the account balance at the end of the firstquarter. Thus, a1 500( ) or the formula for the sum of a Geometric a11 ar1rn S4 n 4, r s account balance at the end of one year is $ ( )4 1 12 Sequences and SeriesDate of 1st Year AdditionsDepositA P 1 nr tn(to the nearest penny)January 1$25 ( )4$ 1$25 ( )3$ 1$25 ( )2$ 1$25 ( )1$ balance on January 1 of following year$ balance at the end of one yearRealWorldApplicationExample6 CommunicatingMathematicsGuided PracticePracticeRead and study the lesson to answer each and contrastarithmetic and Geometric the sequence defined by an ( 3)