tj= =n+6 U7D1 SEQUENCES AND SERIES Introduction A function can be used to generate a sequence of numbers Example: 1(x) = x2 generates

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1 U7D1 SEQUENCES AND SERIES Introduction A function can be used to generate a sequence of numbers Example: 1(x) = x2 generates We have the sequence 1, 4, 9, 16 Thus a sequence is the set of numbers generated by a function, f(x), if x is restricted to the Natural Numbers. Each element in a sequence is referred to as a. We use } t with a to indicate a specific i.e., tj= Types of Sequences 1. Finite Sequences: e.g., 1,4, 9, 16, Infinite Sequences: e.g., 1,4, 9, 16, 25, 36 In general, sequences can be generated using functions that utilize individual or combined mathematical operations, or even previous numbers in the sequence. 1. Arithmetic Sequences: =n+6

2 t..2fl U7D1 HW: Handout f) 4, 7, 10, g) -3, 0, 5, 12,... h) 3, 9, 19, 33, 51, 1234 d 2--,,,,, a) 5, 6, 7, b) 2, 5, 8, c) 1,3, 9, Write the general term for each of the following. c)tk=tkj+k, where t1=5 n2+2n 1. Write the first 3 terms for the following sequences: Examples: tk 2 =tk +tk+1, where t1 =1andl =1 3. Recursive Sequences: 2. Geometric Sequences:

3 a) 3,9,27,. e) f)5,50,500,..... b) 4, 8, 12, c) 9,3,1,... d) 100,95,90, Look at the pattern, then write the next 2 terms in each of the following sequences. b) t,,=t,,1 St,,21wheret1=3andt2= 2,n3 a) t=t,1+t,2,wheret1=1andt2=1,n3 5. Determine the first 6 terms of each of the following sequences. e) -2, -6, -18, -54,... f) c) 2, 4, 8, 16,... d) 2,8,32,128,. a) 2,4,6,8,... b)1,-1,1,-1, Determine a general term for each of the following sequences. c) t,,=6 t11wheret1=3 d)t=(t1)2wheret1= 1 a) t=t1 2,wheret1=5 b)t=2t11wheret1= 5 3. Determine the first 4 terms for each of the following recursive sequences. i),=2n+3 j)=3 g) t=2 h)=1 n3 e) t=3 n f)t= n 1 c)t=1 a) t=3n b)t,=n 2 2. Determine the first 4 terms for each of the following sequences. g) -2, -4, -6,.. Intro Homework U7D1 MCR 3U1 SEQUENCES AND SERIES

4 Answers la) 81, 243 lb) 16, ic) id) 85, 80 3,9 11 le) if) 5000, ig) -8,-b 2a) 3,6,9,12 2b) -1, 0, 1, 2 2c) i!!! 2d)0, 3, 8, e) 2, 1, 0, -1 2f) 2g) 3, 12,27, 48 2h) 0, -7, -26, -63 2i) 5, 7, 9, ii j) 111 3a) 5, 7,,9, ii 3b) -5, -10, -20, -40 3c) 3,3,3,3 3d)-i, 1,1,1 4a), =2n 4b) t =( 1) 4c) t, =2 4d), =221 4e) 1, = 2(3) 4f) t =4- Sa) 1, 1, 2, 3, 5, 8 Sb) 3, -2, 7, 1, 22, 25

5 1. 3,5,7,9,11 What is similar about the following sequences? 10, 15, 23, 29 U702 HW: p. 441 #1-8(eoo 7 note: eoo means every other one in each question- a, c, e, etc.), 9, 4. Describe the arithmetic sequence t,, = 3n 2 as a recursive sequence. 3. Determine t50 if = 5 and th = 26 for an arithmetic sequence. 2. Determine the number of terms in the sequence.... 3, 7, 11, a)2, 6,10,14... b)1o1-191j Determine, and t50 for the following arithmetic sequences: Examples: -1)d In general a, a+d, a+2d, a+3d, a+4d or An arithmetic sequence looks like term. The first term is designated as generated by adding a or to the previous All of these sequences are classified as arithmetic sequences since each term is 3. 20, 17, 14, 11, , 4, 9, 14, 19 Arithmetic Sequences U7D2 SEQUENCES AND SERIES

6 3. 5, -10, 20, -40, 80. by multiplying the previous term by the same amount called the All of these sequences are classified as geometric sequences since each term is generated 2. 2, 10, 50, , 6, 18, 54. U7D3 HW: p.452 #1-7(eoo), 9, 12, 16 recursive sequence. 4. Express the geometric sequences defined by the general term = 3, as a (2 b)if t3=60 and t7=960. a)if t3=15 andt5= Determine if for each of the following geometric sequences: 3, 6, 12, Determine the number of terms in the sequence a)5, 20, 80, b) Determine and 110 for the following geometric sequences: Examples: tn= t=or a, or, or2, Or3 or A geometric sequence looks like In general Geometric Sequences What is similar about the following sequences? U7D3MCR 3UISEQUENCES AND SERIES

7 S3g 54 5n where: If the sequence is t1, 2, 31, then the series is An arithmetic series is the of the terms of an arithmetic sequence. U7D4 HW: p.469 #1-5(eoo) find Findthesumofl If the sum of n terms of a sequence is given by 3, = fl2 + fl 1. Find the sum of the first 100 terms of Examples: [t +t And so we have two different versions of the same formula. S nil [ 2 5 =..[a a+(n 1)d] Sn Or the formula can be written as n/s dig ais 5..[2a+(n1)d] In general 54 1+t2+t3+t4 53 t1+t2+t3 2 =IZ Arithmetic Series U7D4 MCR 3UISEQUENCES AND SERIES

8 2 =t1+t2=a+ar s =t1=o 1, s2 5, 5.Swhere: If the sequence is tj, 2, 31 then the series is A geometric series is the of the terms of a geometric sequence. U7DS HW: p. 452 #1-7(eoo), 9, 12, 16 Geometric (alternate version) Arithmetic Series Arithmetic Arithmetic Sequence General Term Geometric Sequence Summary: Formula List 1. Find the sum of the first 10 terms of 5, 10, 20,40, 2. Find thesum of Examples: n/s p s a/s p i -1),r1 In general 54 =+t2+t3+t4-a+or+ar ar s =t1+t2+t3=a+ar+ar2 Geometric Series U7DS MCR 3UISEQUENCES AND SERIES

9 1. If possible, write out the first few terms of the series. 2. Determine if it is Arithmetic or Geometric, and a Sequence or Series Some hints for solving Story Questions: U7D6 HW: p.470 #11, 13-20, p.477 #7-13 geometric sequence. 7. Determine the value of x such x 2, 2 6. Determine the value of x such x 4, x, x + 10, are consecutive terms of a arithmetic sequence. 2x + 1, 5x + 4, are consecutive terms of an 5. How many multiples of Bare there between 10 and 1000? 4. How many multiples of 5 are from 20 to 200? would you research? 3. Suppose you researched your ancestors back ten generations. How many people stereo after 6 years. 2. A stereo system costing $1200 depreciates by 30 % per year. Find the value of the there are 43 blocks in the first row and 11 blocks in the top row, how many rows high is the wall? 1. A wall of blocks is built up so that each row has 2 less blocks from the previous row. If 4. Solve and write a conclusion. 2. List Given and Unknowns 3. Determine which formula(s) you need to use Sequences & Series Story Questions U7D6MCR 3UISEQUENCES AND SERIES

10 U7D7 MCR 3UIsEQuENcE5 AND SERIES Pascal s Triangle and Binomial Theorem Preamble A binomial is an algebraic expression containing two terms. Ex. 3x+1, 1 x2 3x 7y Today we will learn how to expand binomials raised to any power without the use of tedious and lengthy calculations. Ex. (x 3)2 = (a+b)2 = (3x 2y) = Part A Expand and simplify the following. Place your final answer below. Keep in mind that (x + = (x + y)(x previous answer to proceed. (x+y) = + y), in other words you may use your (x+y) = (x+y)2 (x y)3 (x y)4 (x+y)5 = = = = Part B List all of the patterns you see in the final answers in the expansions. (done the long way)

11 n row sum different Part F How is the expansion of (x from (x + (x+y)8 = Part E Using Pascal s triangle and the patterns you have discovered today, expand Part D List some of the characteristics of the numbers in Pascal s triangle. This is Pascal s Triangle. Then add the coefficients in each row. Part C Write the coefficients of the expansions below centering each row in the space.

12 (x-y)3 (x-y)4 (x-y)5 (x2 )6 (2x 1) 5 Part H Using binomial Theorem expand the following: (a+b) = Binomial Theorem Part H We can generate the coefficients on your calculator if you have a special button. (x2,j)4 (x2_1)6 (2x+3)6 Part C Expand the following.

13 d) (2x l)5 e) (2x+y)4 0 (x 7 a) (x+2)3 b) (y 5 c) (4x+l)5 1. Find the binomial expansion of each expression in simplified for. 8. Write the first three terms and the 7th term in the expansion of (x. Simplify. a) (.& i) b) (x+2) 2 c) (x 2)8 d) (1 2x) 7. Write the first four terms in each expansion. Simplify each term. e) (x 2y)4 0 (2x+3y)3 g) (+1 h) (3x+2y2) 6. Expand and simplif using the binomial theorem. a) (x 2)6 b) (x 3) c) (2 x)5 d) (l+x2)6 have two middle terms? c) When does the expansion of (x + have one middle term? When does it b) Which expansion in part a has a middle term? Which has two middle terms? 5. a) How many terms are there in the expansions of (x + and (x + coefficient of x2y4. 4. In the expansion of (x +, explain why the coefficient of x4y2 is the same a) What is the value n? b) What is the coefficient of x3y4? c) Which term in the expansion is this? 3. In the binomial of (x + a term involving x3y4 occurs. b) What is the coefficient of x5y3? 2. a) Explain how the term x5y3 is formed in the expansion of (x+y)8 Binomial Theorem and Pascal s Triangle WORKSHEET

14 e) 16x4 +32x3y+24x2y2 +Sxy3 +y1 If) x5 35x4 +490x3 3431k x l6807 Ic) 1024x x4 +640x3 +160x2 +20x+1 Id) 32x5 80x4 +80x3 40x2 +lox 1 Ia) x3+6x2+12x+8 ib) y4 20y3+150y2 500y+625 8) x x 1y+ 264x10y x6y6 7c) x8 16x7+112x6 448x5 7d) 1024x 512x9+256x8 128x7... 7a) x5 +1Ox j+45x +12Oxj 7b) x12 +24x x h) 243x5 +810x4y2 -i-1080x3y4 +720x2y6 +24Oy8 +32y 6e) x4 8x3y+24x2y2 32xy3 +16y4 6f) 8x3 +36x2y+54xy2 +27y3 6g) note final answer has a common denominator + 5? + lox6 + lox3 + 5? + I x 6a) x6 +I2x5 +60x4 +160x3 +240x2 +192x+54 6b) x4 12x3 +54x2 108x+81 6c) (x5 lox4 +40x3 81k2 +80x 32) 6d) x 2 +6x +15x8 +20x6 +15x4 +6x2 +1 5c) a single middle term when n is even and 2 middle tenns when n is odd 5a) 16, 17 5b) (x + has a middle term since it has 17 terms, but (x + has 2 middle terms 4)15 3a) 7 3b) 35 3c) 51h term 2a) [8]xsya 2b) 56 Answers: