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31 Pages

### ch.13 Sequences and Series

Course: MATH 2412, Spring 2012
School: Austin CC
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Word Count: 1234

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13 Sequences Chapter and Series 13.1 Sequences Key Points Definition of a sequence Arithmetic and geometric sequences Sequences Examples Finite infinite Notation Notation for Sequences and Examples We denote the terms of a sequence by a1, a2, a3, . . . , an, . . . so that a1 is the first term, a2 is the second term, and so on. We use an to denote the nth or general term of the sequence. If there is a...

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13 Sequences Chapter and Series 13.1 Sequences Key Points Definition of a sequence Arithmetic and geometric sequences Sequences Examples Finite infinite Notation Notation for Sequences and Examples We denote the terms of a sequence by a1, a2, a3, . . . , an, . . . so that a1 is the first term, a2 is the second term, and so on. We use an to denote the nth or general term of the sequence. If there is a pattern in the sequence, we may be able to find a formula for an. Example 1 (a) 0, 1, 4, 9, 16, 25, . . . is the sequence of squares of integers. (b) 2, 4, 8, 16, 32, . . . is the sequence of positive integer powers of 2. (c) 3, 1, 4, 1, 5, 9, . . . is the sequence of digits in the decimal expansion of . (d) 3.9, 5.3, 7.2, 9.6, 12.9, 17.1, 23.1, 38.6, 50.2 is the sequence of U.S. population figures, in millions,for the census reports (1790 -1880). (e) 3.5, 4.2, 5.1, 5.9, 6.7, 8.1, 9.4, 10.6, 10.1, 7.1, 3.8, 2.1, 1.4, 1.1 is the sequence of pager subscribers in Japan, in millions, from the years Functions Modeling 1989 -2002. Change: A Preparation for Calculus, 4th Notation for Sequences and Examples Like Example 3 List the first 5 terms of the sequence an = f(n), where f(x) = 400 20x. Functions Modeling Change: A Preparation for Calculus, 4th Arithmetic Sequences Example 4 Which of the following sequences are arithmetic? (a) 9, 5, 1,3,7 (b) 3, 6, 12, 24, 48 (c) 2, 2 + p, 2 + 2p, 2 + 3p (d) 10, 5, 0, 5, 10 Solution (a) Each term is obtained from the previous term by subtracting 4. This sequence is arithmetic. (b) This sequence is not arithmetic: each terms is twice the previous term. The differences are 3, 6, 12, 24. (c) This sequence is arithmetic: p is added to each term to obtain the next term. (d) This is not arithmetic. The difference between the second and first terms is 5, but the difference between the fifth and fourth terms is 5. Functions Modeling Change: A Preparation for Calculus, 4th Find the 8th term and the nth term of the arithmetic sequences. Arithmetic Sequences For n 1, the nth term of an arithmetic sequence is an = a1 + (n 1)d, where a1 is the first term, and d is the difference between consecutive terms. Functions Modeling Change: A Preparation for Calculus, 4th Geometric Sequences Example 6 Which of the following sequences are geometric? (a) 5, 25, 125, 625, . . . (c) 12, 6, 4, 3, . . . (b) 8, 4,2, 1,1/2 , . . . Solution (a) This sequence is geometric. Each term is 5 times the previous term. Note that the ratio of any term to its predecessor is 5. (b) This sequence is geometric. The ratio of any term to the previous term is 1/2 . (c) This sequence is not geometric. The ratios of successive terms are not constant: a2/a1 = 6/12 = 1/2, but Functions Modeling a3/a2 = 4/6 = 2/3. Change: A Preparation for Calculus, 4th Find the 8th term and the nth term of the geometric sequences. Geometric Sequences For n 1, the nth term of a geometric sequence is an = a1rn1, where a1 is the first term, and r is the ratio of consecutive terms. Functions Modeling Change: A Preparation for Calculus, 4th 28,32 Chapter 13 Sequences and Series 13.2 Defining Functions Using Sums: Arithmetic Series Key Points Arithmetic series as models of growth Summation notation Consider a theater with 40 rows of seats. Suppose that the first row contains 18 seats and the number of seats increases by two per row. What is the total number of seats? Series and Partial Sums The sum of the terms of a sequence is called a series. We write Sn for the sum of the first n terms of the sequence, called the nth partial sum. We see that Sn is a function of n, the number of terms in the partial sum. For the sequence {a1, a2, a3, , an, } , S1 = a1 S2 = a1 + a2 S3 = a1 + a2 + a3 Sn = a1 + a2 + a3+ + aFunctions Modeling Change: n A Preparation for Calculus, 4th General Formula for the Sum of Any Arithmetic Series S n = a1 + a2 + a3 + + an = (a1 + an ) + (a2 + an 1 ) + (a3 + an 2 ) + ( 12 n ) pairs Functions Modeling Change: A Preparation for Calculus, 4th Summation Notation The symbol is used to indicate addition. This symbol, pronounced sigma, is the Greek capital letter for S, which stands for sum. Using this notation, we write n ai to stand for the sum a1 + a2 + + an i =1 The tells us we are adding some numbers. The ai tells us that the numbers we are adding are called a1, a2, and so on. The sum begins with a1 and ends with an because the subscript i (at the bottom of the sign) begins at 1 and ends at n (at the top of the sign). Functions Modeling Change: A Preparation for Calculus, 4th Using the Sigma Notation 6,10, 14, 20, 24, 28, 34 Functions Modeling Change: A Preparation for Calculus, 4th Chapter 13 Sequences and Series 13.3 Finite Geometric Series Key Points Geometric series as models of growth Bank Balance A person deposits \$2000 every year in an IRA that pays 6% interest per year, compounded annually. After the first deposit (but before any interest has been earned), the balance in the account in dollars is B1 = 2000. After 1 year has passed, the first deposit has earned interest, so the balance becomes 2000(1.06) dollars. Then the second deposit is made and the balance becomes B2 = 2nd deposit + 1st deposit with interest = 2000 + 2000(1.06) dollars. After 2 years have passed, the third deposit is made, and the balance is B3 = 3rd dep+2nd dep with 1 yr interest +1st dep with 2 yrs interest = 2000 + 2000(1.06)+ 2000(1.06)2. Let Bn be the balance in dollars after n deposits. Then we see that Bn = 2000 + 2000(1.06)+ 2000(1.06)2 + + 2000(1.06)n1. Functions Modeling Change: A Preparation for Calculus, 4th Geometric Series A finite geometric series is a sum of the form Sn = a + ar + ar2 + + arn1 or using sigma notation with first term a and common ratio r: n S n = ar i 1. i =1 Functions Modeling Change: A Preparation for Calculus, 4th The Sum of a Geometric Series The The sum of a finite geometric series of n terms is given by a (1 r n ) 2 n 1 S n = a + ar + ar + + ar = for r 1. (1 r ) Functions Modeling Change: A Preparation for Calculus, 4th The Sum of a Geometric Series Like Example 2 Use the formula for the sum of a geometric series to solve Example 1 with the bank deposits (find B7 and B30). Functions Modeling Change: A Preparation for Calculus, 4th 12,14,6, 4,18,2 Chapter 13 Sequences and Series 13.4 Infinite Geometric Series Key Points Infinite geometric series Long-term Drug Level in the Body In the drug dosage example, what happens to the drug level in the patients body? To find out, we calculate the drug level after 10, 15, 20, and 25 injections. Using the formula based on 20mg of a drug, with a metabolic halflife of one day, injected each day into a person, the amount of the drug present after n days was found to be Qn = 20 (1 ( )n) / (1 ). After the 10th, 15th, 20th, and 25th injection, the drug level in the patients body is given by Q10 = 20 (1 ( )10) / (1 ) = 39.960938 mg, Q15 = 20 (1 ( )15) / (1 ) = 39.998779 mg, Q20 = 20 (1 ( )20) / (1 ) = 39.999962 mg, Q25 = 20 (1 ( )25) / (1 ) = 39.999999 mg. The drug level appears to approach 40 mg Functions Modeling Change: A Preparation for Calculus, 4th The Sum of an Infinite Geometric Series Consider the geometric series Sn = a + ar + ar2 + + arn1. In general, if |r| < 1, then rn 0 as n , so a (1 r n ) a (1 0) a Sn = = as n . 1 r 1 r 1 r Thus, if |r| < 1, the partial sums Sn approach a finite value, S, as n . In this case, we say that the series converges to S. For |r| < 1, the sum of the infinite geometric series is given by For a 2 n 1 i S = a + ar + ar + + ar + = ar = . 1 r i =0 For |r| 1, the value of Sn does not approach a fixed number as n Functions , so we say that the infinite series does not converge. Modeling Change: A Preparation for Calculus, 4th Infinite Geometric Series 8,10,14,16,22
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