Sequences And Series PDF

SEQUENCE AND SERIES BASIC MATHEMATICS (FADS 1004) © 2019, University of Cyberjaya. Please do not reproduce, redistribute or share without the prior express permission of the author. SEQUENCES AND SERIES Sequences and Series Arithmetic Sequences and Partial Sum Geometric Se...

SEQUENCE AND SERIES BASIC MATHEMATICS (FADS 1004) © 2019, University of Cyberjaya. Please do not reproduce, redistribute or share without the prior express permission of the author. SEQUENCES AND SERIES Sequences and Series Arithmetic Sequences and Partial Sum Geometric Sequences and Series Binomial Theorem  Learning Outcomes: At the end of this lesson, student should be able to: ✓ Use sequence notation to write the terms of sequences. ✓ Use factorial notation. ✓ Use summation notation to write sums. ✓ Find the sums of series. ✓ Use sequences and series to model and solve real- life problems. / [email protected] 3  Sequences / [email protected] 4 Example 1 – Writing the Terms of a Sequence Write the first four terms of the sequences given by a. an = 3n – 2 b. an = 3 + (– 1)n. Solution: a. The first four terms of the sequence given by an = 3n – 2 are a1 = 3(1) – 2 = 1 a2 = 3(2) – 2 = 4 1st term a3 = 3(3) – 2 = 7 2nd term a4 = 3(4) – 2 = 10. 3rd term 4th term / [email protected] 5 Example 1 – Solution cont’d b. The first four terms of the sequence given by an = 3 + (–1)n are a1 = 3 + (–1)1 = 3 – 1 = 2 1st term a2 = 3 + (–1)2 = 3+1=4 a3 = 3 + (–1)3 = 3 – 1 = 2 2nd term a4 = 3 + (–1)4 = 3 + 1 = 4. 3rd term 4th term / [email protected] 6 Sequences Some sequences are defined recursively. To define a sequence recursively, you need to be given one or more of the first few terms. All other terms of the sequence are then defined using previous terms. A well-known recursive sequence is the Fibonacci sequence. A sequence is a function where domain is a set of positive integers. FM 1024 / [email protected] 8 Recursive form The n term of a sequence involves one or more of the term preceding it. A recursive formula always has two parts: 1. the starting value for a1. 2. the recursion equation for an as a function of an-1 (the term before it.) / [email protected] 9 Example: Write the first four terms of the sequence: In recursive formulas, each term is used to produce the next term. Follow the movement of the terms through the set up at the left. Answer: -4, 1, 6, 11 16 / [email protected] Fibonacci Sequences The Fibonacci numbers are the sequence of numbers {Fn }n=1 defined by the linear recurrence equation Fn = Fn−1 + Fn−2 (1) with F1 = F2 =1 Leonardo Fibonacci c1175- 1250. As a result of the definition (1), it is conventional to define F0 = 0 The Fibonacci numbers for n = 1, 2,... are 1, 1, 2, 3, 5, 8, 13, 21, … FM 1024 / [email protected] 1 1  Here is a longer list. 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811,... Fibonacci Can you figure out the next following numbers? Sequences FM 1024 / [email protected] 10  Factorial ! / [email protected] 17 Factorial Notation Here are some values of n! for the first several nonnegative integers. Notice that 0! is 1 by definition. 0! = 1 1! = 1 2! = 1 🞄 2 = 2 3! = 1 🞄 2 🞄 3 = 6 4! = 1 🞄 2 🞄 3 🞄 4 = 24 5! = 1 🞄 2 🞄 3 🞄 4 🞄 5 = 120 The value of n does not have to be very large before the value of n! becomes extremely large. For instance, 10! = 3,628,800. / [email protected] 18  Factorial Notation Factorials follow the same conventions for order of operations as do exponents. For instance, 2(n!) = 2(1 2 🞄 3 🞄 4... N) whereas (2n)! = 1 🞄 2 🞄 3 🞄 4... 2n. / [email protected] 19 Example – Writing the Terms of a Sequence Involving Factorials Write the first five terms of the sequence given by n = 0,1,2,3,… Solution: 0th term / [email protected] 20 Example 5 – Solution cont’d 1st term 3rd term 4th term 2nd term / [email protected] 21 Summation Notation There is a convenient notation for the sum of the terms of a finite sequence. It is called summation notation or sigma / [email protected] 23 Example 7 – Summation Notation for a Sum Find each sum. a. b. c. Solution: a. = 3(1) + 3(2) + 3(3) + 3(4) + 3(5) = 3(1 + 2 + 3 + 4 + 5) = 3(15) = 45 / [email protected] 24 Example 7 – Solution cont’d b. = (1 + 32) + (1 + 42) + (1 + 52) + (1 + 62) = 10 + 17 + 26 + 37 = 90 c. / [email protected] 25  In Example 7, note that the lower limit of a summation does not have to be 1. Summation Also note that the index of summation does not have to be the letter i. For instance, in part (b), the Notation letter i is the index of summation. / [email protected] 27 Series / [email protected] 28 Series FM 1024 / [email protected] 29 Example 8 – Finding the Sum of a Series For the series find (a) the third partial sum and (b) the sum. Solution: a. The third partial sum is = 0.3 + 0.03 + 0.003 = 0.333. FM 1024 / [email protected] 30 Example 8 – Solution cont’d b. The sum of the series is = 0.3 + 0.03 + 0.003 + 0.0003 + 0.00003 +... = 0.33333... FM 1024 / [email protected] 31  SEQUENCES AND SERIES Sequences and Series Arithmetic Sequences and Partial Sum Geometric Sequences and Series Binomial Theorem / [email protected] Arithmetic Sequences FM 1024 / [email protected] 34 Example 1 – Examples of Arithmetic Sequences a. The sequence whose nth term is 4n + 3 is arithmetic. For this sequence, the common difference between consecutive terms is 4. 7, 11, 15, 19,... , 4n + 3,... Begin with n = 1. 11 – 7 = 4 b. The sequence whose nth term is 7 – 5n is arithmetic. For this sequence, the common difference between consecutive terms is –5. 2, –3, –8, –13,... , 7 – 5n,... Begin with n = 1. –3 – 2 = –5 FM 1024 / [email protected] 35 Example 1 – Examples of Arithmetic Sequences cont’d c. The sequence whose nth term is is arithmetic. For this sequence, the common difference between consecutive terms is Begin with n = 1. FM 1024 / [email protected] 36 Arithmetic Sequences The sequence 1, 4, 9, 16,... , whose n th term is n2, is not arithmetic. The difference between the first two terms is a2 – a1 = 4 – 1 = 3 but the difference between the second and third terms is a3 – a2 = 9 – 4 = 5. FM 1024 / [email protected] 37 FM 1024 / [email protected] 38  Example 2 – Finding the nth Term of an Arithmetic Sequence Find a formula for the nth term of the arithmetic sequence whose common difference is 3 and whose first term is 2. Solution: You know that the formula for the nth term is of the form an = a1 + ( n – 1)d. Moreover, because the common difference is d = 3 and the first term is a1 = 2, the formula must have the form an = 2 + 3(n – 1). Substitute 2 for a 1 and 3 for d. FM 1024 / [email protected] 39 Example 2 – Solution cont’d So, the formula for the nth term is an = 3n – 1. The sequence therefore has the following form. 2, 5, 8, 11, 14,... , 3n – 1,... FM 1024 / [email protected] 40 Arithmetic Sequences If you know the nth term of an arithmetic sequence and you know the common difference of the sequence, you can find the (n + 1)th term by using the recursion formula an + 1 = an + d. Recursion formula With this formula, you can find any term of an arithmetic sequence, provided that you know the preceding term. For instance, if you know the first term, you can find the second term. Then, knowing the second term, you can find the third term, and so on. FM 1024 / [email protected] 41  The Sum of a Finite Arithmetic Sequence 42 The Sum of a Finite Arithmetic Sequence There is a simple formula for the sum of a finite arithmetic sequence. FM 1024 / [email protected] 43 Example 5 – Finding the Sum of a Finite Arithmetic Sequence Find the sum: 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19. Solution: To begin, notice that the sequence is arithmetic (with a common difference of 2). Moreover, the sequence has 10 terms. So, the sum of the sequence is Sn = (a1 + an) Formula for the sum of an arithmetic sequence FM 1024 / [email protected] 44 Example 5 – Solution cont’d = (1 + 19) Substitute 10 for n, 1 for a1, and 19 for an. = 5(20) Simplify. = 100. FM 1024 / [email protected] 45 The Sum of a Finite Arithmetic Sequence The sum of the first n terms of an infinite sequence is the n th partial sum. The n th partial sum can be found by using the formula for the sum of a finite arithmetic sequence. FM 1024 / [email protected] 46 Geometric Sequence At the end of this lesson, student should be able to; Find the common ratio for a geometric sequence Write n-term of geometric sequence Find the general term of a geometric sequences Find the first and terms of a geometric sequence Find the sum of an infinite geometric series FM 1024 / [email protected] 48 Geometric Sequences A geometric sequence may be thought of as an exponential function whose domain is the set of natural numbers. FM 1024 / [email protected] 48 Geometric Sequences If you know the nth term of a geometric sequence, you can find the (n + 1)th term by multiplying by r. That is, an + 1 = anr. FM 1024 / [email protected] 49 Example 4 – Finding a Term of a Geometric Sequence Find the 12th term of the geometric sequence 5, 15, 45,.... Solution: The common ratio of this sequence is Because the first term is a1 = 5, you can determine the 12th term (n = 12) to be an = a1r n – 1 Formula for geometric sequence FM 1024 / [email protected] 50 Example 4 – Solution cont’d a12 = 5(3)12 – 1 Substitute 5 for a1, 3 for r, and 12 for n. = 5(177,147) Use a calculator. Simplify. = 885,735. FM 1024 / [email protected] 51 Geometric Sequences If you know any two terms of a geometric sequence, you can use that information to find a formula for the nth term of the sequence. FM 1024 / [email protected] 52 The Sum of a Finite Geometric Sequence The formula for the sum of a finite geometric sequence is as follows. FM 1024 / [email protected] 54  Geometric Series The summation of the terms of an infinite geometric sequence is called an infinite geometric series or simply a geometric series. The formula for the sum of a finite geometric sequence can, depending on the value of r, be extended to produce a formula for the sum of an infinite geometric series. Specifically, if the common ratio r has the property that | r |  1, it can be shown that r n becomes arbitrarily close to zero as n increases without bound. FM 1024 / [email protected] 60 Geometric Series Consequently, This result is summarized as follows. Note that if | r |  1, the series does not have a sum. FM 1024 / [email protected] 61 Example 7 – Finding the Sum of an Infinite Geometric Series Find each sum. a. Solution: a. FM 1024 / [email protected] 62  Sequences and Series SEQUENCES Arithmetic Sequences and Partial Sum AND SERIES Geometric Sequences and Series Binomial Theorem At the end of this lesson, student should be able to; Recognize patterns in binomial expansions Evaluate a binomial coefficient Expand a binomial raised to a power Find a particular term in a binomial expansion FM 1024 / [email protected] 65 Pattern in Binomial Expansions (a + b)1 = a + b (a + b)2 = a2 + 2ab + b2 (a + b)3 = a3 + 3a2b + 3ab2 + b3 (a + b)4 = a4 + 4a3b + 6a2b2 + 4ab3 + b4 (a + b)5 = a5 + 5a4b +10a3b2 +10a2b3 + 5ab4 + b5 FM 1024 / [email protected] 66 Pascal Triangle 1 1 11 2 1 1 3 3 1 1 4 6 4 1 5 10 10 5 1 1 6 15 20 15 6 1 (a + b)6 = a6 + 6a5b +15a4b2 + 20a3b3 +15a2b4 + 6ab5 + b6 FM 1024 / [email protected] 67 Binomial Coefficient n r    Definition: for non negative integers n and r , with n greater and equal than r, the expression is n n! r  =   r !(n − r)! FM 1024 / [email protected] 68 The Binomial Theorem The formal expression of the Binomial Theorem n  n  n−k k (a + b) =    a b n k =0   k Binomial expansion n n n n n (a + b)n =   an +   an−1b +   an−2b2 +   an−3b3 +... +   bn 0 1   2 3  n FM 1024 / [email protected] 69 Example; Use Binomial Theorem to expand (x + 2)4 Using Binomial Expansion  4  4  4  4  4 (x + 2)4 =   (x) 4−0 (2)0 +  (x) 4−1 (2)1 +  (x) 4−2 (2)2 +  (x) 4−3 (2)3 +  (x) 4−4 (2)4 0 1  2  3 4 = (1)x4 (1) + (4)x3 (2) + (6)x 2 (4) + (4)x1 (8) + (1)x0 (16) = x 4 + 8x 3 + 24x 2 + 32x +16 FM 1024 / [email protected] 70 Expand (2x − 5 y)5 (2x − 5 y)5  5  5  5 =   (2x) (−5 y) +   (2x) (−5 y) +  (2x)5−2 (−5 y)2 + 5−0 0 5−1 1 0 1  2  5  5  5  3 (2x) 5−3 (−5 y) 3 +  4 (2x) 5−4 (−5 y) 4 +  5 (2x) 5−5 (−5 y)5       = (1)(32x5 )(1) + (5)(16x4 )(−5 y) + (10)(8x3 )(25y2 ) + (10)(4x2 )(−125 y3 ) + (5)(2x)(625 y4 ) + (1)(1)(−3125 y5 ) = 32x5 − 400x4 y + 2000x3 y2 − 5000x2 y3 + 6250xy4 − 3125y5 FM 1024 / [email protected] 71 Finding particular term in a Binomial expansion The (r+1) term of the expansion of is (a + b)n 𝑇𝑟+1 =  n  n−r r r a b   FM 1024 / [email protected] 72 Example; i) Find the forth term in the expansion of ii) Find the tenth term in the expansion FM 1024 / [email protected] 73  (3x + 2 y)7 𝑇𝑟+1 =  n  n−r r 4th term r a b   7 𝑇4 = 𝑇3+1 =   (3x)7−3 (2 y)3 = (35)(3x)4 (2 y)3 = 22680x4 y3 3 (x + 3)12 Tenth term 12  𝑇10 = 𝑇9+1 =   (x) 3 (3)9 = (220)x3 (19683) = 4330260x3 9 FM 1024 / [email protected] 74 Thank you Address Telephone Website University of Cyberjaya 03 - 8313 7000 www.cyberjaya.edu.my Persiaran Bestari, Cyber 11, 63000 Cyberjaya, Facsimile Email Selangor Darul Ehsan, Malaysia. 03 – 8313 7001 [email protected]