Pre-Calculus. Slide 1 / 145. Slide 2 / 145. Slide 3 / 145. Sequences and Series. Table of Contents

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Slide 1 / 145 Pre-Calculus Slide 2 / 145 Sequences and Series 2015-03-24 www.njctl.

1 Slide 1 / 145 Pre-Calculus Slide 2 / 145 Sequences and Series Table of Contents s Arithmetic Series Geometric Sequences Geometric Series Infinite Geometric Series Special Sequences Binomial Theorem click on the topic to go to that section Slide 3 / 145

2 Slide 4 / 145 Arithmetic Sequences Return to Table of Contents Slide 5 / 145 Slide 6 / 145

3 Slide 7 / Find the next term in the arithmetic sequence: 3, 9, 15, 21,... Slide 8 / Find the next term in the arithmetic sequence: -8, -4, 0, 4,... Slide 9 / Find the next term in the arithmetic sequence: 2.3, 4.5, 6.7, 8.9,...

4 Slide 10 / Find the value of d in the arithmetic sequence: 10, -2, -14, -26,... Slide 11 / Find the value of d in the arithmetic sequence: -8, 3, 14, 25,... Slide 12 / 145 As we study sequences we need a way of naming the terms. a 1 to represent the first term, a 2 to represent the second term, a 3 to represent the third term, and so on in this manner. If we were talking about the 8 th term we would use a 8. When we want to talk about general term call it the n th term and use a n.

5 Slide 13 / 145 Write the first four terms of the arithmetic sequence that is described. a 1 = 4; d = 6 a 1 = 3; d = -3 a 1 = 0.5; d = 2.3 a 2 = 7; d = 5 Slide 14 / Which sequence matches the description? A 4, 6, 8, 10 B 2, 6,10, 14 C 2, 8, 32, 128 D 4, 8, 16, 32 Slide 15 / Which sequence matches the description? A -3, -7, -10, -14 B -4, -7, -11, -13 C -3, -7, -11, -15 D -3, 1, 5, 9

6 Slide 16 / Which sequence matches the description? A 7, 10, 13, 16 B 4, 7, 10, 13 C 1, 4, 7,10 D 3, 5, 7, 9 Slide 17 / 145 To find a specific term,say the 5 th or a 5, you could write out all of the terms. But what about the 100 th term(or a 100)? We need to find a formula to get there directly without writing out the whole list. Consider: 3, 9, 15, 21, 27, 33, 39,... Slide 18 / 145 a1 3 a2 9 = 3+6 a3 15 = 3+12 = 3+2(6) a4 21 = 3+18 = 3+3(6) a5 27 = 3+24 = 3+ 4(6) Do you see a pattern that relates the term number to its value? a6 33 = 3+30 = 3+5(6) a7 39 = 3+36 = 3+6(6) click

7 Slide 19 / 145 Example Find the 21 st term of the arithmetic sequence with a 1 = 4 and d = 3. Example Find the 12 th term of the arithmetic sequence with a 1 = 6 and d = -5. Slide 20 / 145 Example Find the 1 st term of the arithmetic sequence with a 15 = 30 and d = 7. Example Find the 1 st term of the arithmetic sequence with a 17 = 4 and d = -2. Slide 21 / 145 Example Find d of the arithmetic sequence with a 15 = 42 and a 1=3. Example Find the term number n of the arithmetic sequence with a n = 6, a 1=-34 and d = 4.

8 Slide 22 / Find a 11 when a 1 = 13 and d = 6. Slide 23 / Find a 17 when a 1 = 12 and d = -0.5 Slide 24 / Find a 17 for the sequence 2, 4.5, 7, 9.5,...

9 Slide 25 / Find the common difference d when a 1 = 12 and a 13= 6. Slide 26 / Find n such a 1 = 12, a n= -20, and d = -2. Slide 27 / Tom works at a car dealership selling cars. He is paid $4000 a month plus a $300 commission for every car he sells after the first car. How much did he make in April if he sold 14 cars?

10 Find the missing terms in the arithmetic sequence. 4, 6, 8, 10, 5, 10,, 20, 12, 9, 6 6,, 14 Slide 28 / 145 Notice in the last example d was added to get from 6 to and another d was added to get from to 14. Or 6 + 2d = 14 Find the missing terms 2,,, 23 Slide 29 / 145 4,,, -14 7,,,, 39-9,,,,, Find the missing term: 4,, -16 A -20 B -10 C -6 D 2 Slide 30 / 145

11 Slide 31 / Find the missing terms: -10,,, 8 A -6, -2 B -6, 2 C -5, 1 D -4, 2 Slide 32 / Find the missing terms: 12,,, 75 A 27, 51 B 33, 54 C 37, 51 D 34, 58 Slide 33 / Find d for the arithmetic sequence: 5,,,, 21

12 Slide 34 / 145 Arithmetic Series Return to Table of Contents Arithmetic Series Slide 35 / 145 Arithmetic Series An arithmetic series is the sum of the terms in the arithmetic sequence. S n represents the sum of the first n terms. Consider 4, 7,10, 13, 16, 19, 22,... S 4= = 34 S 6= = 69 Arithmetic Series Slide 36 / Consider 3, 9, 15, 21, 27, 33, 39,... what is S 4?

13 Arithmetic Series 20 Consider 3, 9, 15, 21, 27, 33, 39,... what is S 5? Slide 37 / 145 Arithmetic Series 21 Consider 3, 9, 15, 21, 27, 33, 39,... what is S 7? Slide 38 / 145 Arithmetic Series Slide 39 / 145 Arithmetic Series Suppose we wanted to find the first 100 terms of 4, 7,10, 13, 16, 19, 22,... or S 100? There must be a short cut. The 100 th term of the sequence is 301 using a 100 = 4 + (100-1)3. S 100 = If we add the smallest and largest ( )= 305 If we add the next two ( ) = 305 and continue ( ) until they are all paired up ( ). We now have 50 pairs of 305, so... S 100 = = 50(305) = Do you see a pattern?

14 Slide 40 / 145 Arithmetic Series Slide 41 / 145 Find S n for each arithmetic series. Example: a 1 = 7 and a 12 = -23 Example: a 1 = 6, n = 10, and d = 9 Arithmetic Series Slide 42 / 145 Find S n for each arithmetic series. Example: a 12= 30 and d = -7 Example: a 1 = 2, a n = 32, and d = 5

15 Arithmetic Series 22 Find the S n for the arithmetic series described: a 1 = 19 and a 12 = 37. Slide 43 / 145 Arithmetic Series 23 Find the S n for the arithmetic series described: a 1 = 30 and a 17 = -45. Slide 44 / 145 Arithmetic Series 24 Find the S n for the arithmetic series described: a 1 = 20, n = 8, and d = 6. Slide 45 / 145

16 Arithmetic Series Slide 46 / Find the S n for the arithmetic series described: a n = 20, n = 9, and d = -4. Arithmetic Series Slide 47 / Find the S n for the arithmetic series described: Arithmetic Series Slide 48 / 145 Sigma Notation Sigma ( ) is the Greek letter S. And means the sum of the terms in a sequence. The difference between S n and sigma is that S n is always the first to the n th term. means start with 3 rd and sum up through the 9 th term.

17 Slide 49 / 145 Arithmetic Series Slide 50 / Evaluate Arithmetic Series Slide 51 / Evaluate

Arithmetic Series Slide 52 / 145 29 Evaluate Arithmetic Series Slide

18 Arithmetic Series Slide 52 / Evaluate Arithmetic Series Slide 53 / 145 How would evaluate? Arithmetic Series Slide 54 / Evaluate

19 Slide 55 / 145 Geometric Sequences Return to Table of Contents Slide 56 / 145 Geometric Sequence Slide 57 / Find the next term in geometric sequence: 6, -12, 24, -48, 96,...

20 Geometric Sequence Slide 58 / Find the next term in geometric sequence: 64, 16, 4, 1,... Geometric Sequence Slide 59 / Find the next term in geometric sequence: 6, 15, 37.5, 93.75,... Geometric Sequence 34 Is the following sequence a geomtric one? 48, 24, 12, 8, 4, 2, 1 Slide 60 / 145 Yes No

21 Geometric Sequence Geometric Sequence Geometric Sequences can be described by giving the first term, a 1, and the common ratio, r. Examples: Find the first five terms of the geometric sequence described. 1) a 1 = 6 and r = 3 2) a 1 = 8 and r = -.5 3) a 1 = -24 and r = 1.5 Slide 61 / 145 4) a 1 = 12 and r = 2 / 3 Geometric Sequence 35 Find the first four terms of the geometric sequence described: a 1 = 6 and r = 4. A 6, 24, 96, 384 B 4, 24, 144, 864 C 6, 10, 14, 18 D 4, 10, 16, 22 Slide 62 / 145 Geometric Sequence 36 Find the first four terms of the geometric sequence described: a 1 = 12 and r = - 1 / 2. A 12, -6, 3, -.75 B 12, -6, 3, -1.5 C 6, -3, 1.5, -.75 D -6, 3, -1.5,.75 Slide 63 / 145

Geometric Sequence Slide 64 / 145 37 Find the first four terms of the geometric sequence described: a 1 = 7 and r = -2.

22 Geometric Sequence Slide 64 / Find the first four terms of the geometric sequence described: a 1 = 7 and r = -2. A 14, 28, 56, 112 B -14, 28, -56, 112 C 7, -14, 28, -56 D -7, 14, -28, 56 Geometric Sequence Slide 65 / 145 Geometric Sequence Consider the sequence: 3, 6, 12, 24, 48, 96,... To find the seventh term, just multiply the sixth term by 2. But what if I want to find the 20 th term? Look for a pattern: a1 3 a2 6 = 3(2) a3 12 = 3(4) = 3(2) 2 Do you see a pattern? a4 24 = 3(8) = 3(2) 3 a5 48 = 3(16) = 3(2) 4 click a6 96 = 3(32) = 3(2) 5 a7 192 = 3(64) = 3(2) 6 Geometric Sequence Slide 66 / 145 Geometric Sequence

23 Geometric Sequence Geometric Sequence Find the indicated term. Example: a 20 given a 1 =3 and r = 2. Slide 67 / 145 Example: a 10 for 2187, 729, 243, 81 Geometric Sequence Geometric Sequence Example: Find r if a 6 =.2 and a 1 = 625 Slide 68 / 145 Example: Find n if a 1 = 6, a n = 98,304 and r = 4. Geometric Sequence 38 Find a 12 in a geometric sequence where a 1 = 5 and r = 3. Slide 69 / 145

24 Geometric Sequence Slide 70 / Find a 10 in a geometric sequence where a 1 = 7 and r = -2. Geometric Sequence Slide 71 / Find a 7 in a geometric sequence where a 1 = 10 and r = - 1 / 2. Geometric Sequence Slide 72 / Find r of a geometric sequence where a 1 = 3 and a 10=59049.

25 Geometric Sequence Slide 73 / Find n of a geometric sequence where a 1 = 72, r =.5, and a n = 2.25 Geometric Sequence Geometric Sequence Find the missing term in the geometric sequence 3, 9, 27, 5, 1, 1 / 5,, -10, 50, ,, -32 Slide 74 / 145 Geometric Sequence Geometric Sequence Find the missing terms in the geometric sequence. Slide 75 / 145 5,,, 40-54,,, 16 4,,,, ,,,,, 4.5

26 Geometric Sequence Slide 76 / What number(s) fill in the blanks of the geometric sequence:, 14, 98, 686 A -14 B -7 C -2 D 2 E 7 F 8 G 10 H 12 I 28 J 50 Slide 77 / 145 Slide 78 / 145

27 Slide 79 / 145 Geometric Series Return to Table of Contents Geometric Series Slide 80 / 145 The sum of a geometric series can be found using the formula: Examples: Find S n for each example. 1) a 1= 5, r= 3, n= 6 2) a 1= -3, r= -2, n=7 Geometric Series Slide 81 / 145 Sometimes information will be missing, so that using isn't possible to start. Look to use information. Example: a 1 = 16 and a 5 = 243, find S 5 to find missing

28 Geometric Series Slide 82 / 145 Example: a 1 = 16 and a 5 = 243, find S 5 (continued) Geometric Series Slide 83 / Find the indicated sum of the geometric series described: a 1 = 10, n = 6, and r = 6 Geometric Series Slide 84 / Find the indicated sum of the geometric series described: a 1 = -2, n = 5, and r = 1 / 4

29 Geometric Series Slide 85 / Find the indicated sum of the geometric series described: a 1 = 8, n = 6, and r = -2 Geometric Series Slide 86 / Find the indicated sum of the geometric series described: a 1 = 8, n = 5, and a 6 = 8192 Geometric Series Slide 87 / Find the indicated sum of the geometric series described: r = 6, n = 4, and a 4 = 2592

30 Geometric Series Slide 88 / Find the indicated sum of the geometric series described: find S 7 Geometric Series Slide 89 / 145 Sigma ( )can be used to describe the sum of a geometric series. We can still use the sigma notation., but to do so we must examine Examples: n = 4 Why? The bounds on below and on top indicate that. a 1 = 6 Why? The coefficient is all that remains when the base is powered by 0. r = 3 Why? In the exponential chapter this was our growth rate. Geometric Series Slide 90 / Find the sum:

31 Geometric Series Slide 91 / Find the sum: Geometric Series 54 Find the sum: Slide 92 / 145 Slide 93 / 145 Infinite Geometric Series Return to Table of Contents

32 Infinite Geometric Series Slide 94 / 145 n an n Sn , ,561 Because r = 3, the series is growing at increasing rate , , ,840 Infinite Geometric Series Slide 95 / 145 n an Because r = 1 / 2, a n --> 0. Notice S n --> an asymptote? n Sn Infinite Geometric Series Slide 96 / 145 n an Because r = - 1 / 2, a n --> 0. Notice S n --> an asymptote? n Sn

Example 2: a 1 = 64 and r= 1 / 2 Example 3: a 1 = 64 and r= - 1 / 2 Infinite Geometric

33 Infinite Geometric Series Slide 97 / 145 For an infinite geometric series to approach a value, -1 < r < 1 then The examples from the previous slides: Example 1: a 1 = 3 and r = 3. Example 2: a 1 = 64 and r= 1 / 2 Example 3: a 1 = 64 and r= - 1 / 2 Infinite Geometric Series Slide 98 / Find the sum of this infinite geometric series, if one exists: Slide 99 / 145

34 Slide 100 / 145 Slide 101 / 145 Infinite Geometric Series Slide 102 / Find the sum of this infinite geometric series, if one exists:

35 Infinite Geometric Series Slide 103 / Find the sum of this infinite geometric series, if one exists: Slide 104 / 145 Special Sequences Return to Table of Contents Special Sequences Slide 105 / 145 A recursive formula is one in which to find a term you need to know the preceding term. So to know term 8 you need the value of term 7, and to know the n th term you need term n-1 In each example, find the first 5 terms a1 = 6, an = an-1 +7 a1 =10, an = 4an-1 a1 = 12, an = 2an a1 10 a a2 40 a a3 160 a a4 640 a a a 5 237

36 Special Sequences Slide 106 / Find the first four terms of the sequence: a 1 = 6 and a n = a n-1-3 A 6, 3, 0, -3 B 6, -18, 54, -162 C -3, 3, 9, 15 D -3, 18, 108, 648 Special Sequences Slide 107 / Find the first four terms of the sequence: a 1 = 6 and a n = -3a n-1 A 6, 3, 0, -3 B 6, -18, 54, -162 C -3, 3, 9, 15 D -3, 18, 108, 648 Special Sequences Slide 108 / Find the first four terms of the sequence: a 1 = 6 and a n = -3a n A 6, -22, 70, -216 B 6, -22, 70, -214 C 6, -14, 46, -134 D 6, -14, 46, -142

Special Sequences Slide 109 / 145 a1 = 6, an = an-1 +7 a1 =10, an = 4an-1 a1 = 12, an = 2an-1 +3 6 a1 10 a 1 12 13 a2 40 a 2 27 20 a3 160 a 3 57 27 a4 640 a 4 117 34 a5 2560 a 5 237 The recursive

37 Special Sequences Slide 109 / 145 a1 = 6, an = an-1 +7 a1 =10, an = 4an-1 a1 = 12, an = 2an a1 10 a a2 40 a a3 160 a a4 640 a a a The recursive formula in the first column represents an. We can write this formula so that we find a n directly. Recall: We will need a 1 and d,they can be found both from the table and the recursive formula. Special Sequences Slide 110 / 145 a1 = 6, an = an-1 +7 a1 =10, an = 4an-1 a1 = 12, an = 2an a1 10 a a2 40 a a3 160 a a4 640 a a a The recursive formula in the second column represents a Geometric Sequence. We can write this formula so that we find a n directly. Recall: We will need a 1 and r,they can be found both from the table and the recursive formula. Special Sequences Slide 111 / 145 a1 = 6, an = an-1 +7 a1 =10, an = 4an-1 a1 = 12, an = 2an a1 10 a a2 40 a a3 160 a a4 640 a a a The recursive formula in the third column represents neither an Arithmetic or Geometric Sequence. This observation comes from the formula where you have both multiply and add from one term to the next.

38 Special Sequences Slide 112 / Identify the sequence as arithmetic, geometric,or neither. a 1 = 12, a n = 2a n-1 +7 A B C Arithmetic Geometric Neither Special Sequences Slide 113 / Identify the sequence as arithmetic, geometric,or neither. a 1 = 20, a n = 5a n-1 A B C Arithmetic Geometric Neither Special Sequences Slide 114 / Which equation could be used to find the n th term of the recursive formula directly? a 1 = 20, a n = 5a n-1 A B C D a n = 20 + (n-1)5 a n = 20(5) n-1 a n = 5 + (n-1)20 a n = 5(20) n-1

39 Special Sequences Slide 115 / Identify the sequence as arithmetic, geometric,or neither. a 1 = -12, a n = a n-1-8 A B C Arithmetic Geometric Neither Special Sequences Slide 116 / Which equation could be used to find the n th term of the recursive formula directly? a 1 = -12, a n = a n-1-8 A B C D a n = (n-1)(-8) a n = -12(-8) n-1 a n = -8 + (n-1)(-12) a n = -8(-12) n-1 Special Sequences Slide 117 / Identify the sequence as arithmetic, geometric,or neither. a 1 = 10, a n = a n A B C Arithmetic Geometric Neither

40 Special Sequences Slide 118 / Which equation could be used to find the n th term of the recursive formula directly? a 1 = 10, a n = a n A B C D a n = 10 + (n-1)(8) a n = 10(8) n-1 a n = 8 + (n-1)(10) a n = 8(10) n-1 Special Sequences Slide 119 / Identify the sequence as arithmetic, geometric,or neither. a 1 = 24, a n = ( 1 / 2)a n-1 A B C Arithmetic Geometric Neither Special Sequences Slide 120 / Which equation could be used to find the n th term of the recursive formula directly? a 1 = 24, a n = ( 1 / 2)a n-1 A B C D a n = 24 + (n-1)( 1 / 2) a n = 24( 1 / 2) n-1 a n = ( 1 / 2) + (n-1)24 a n = ( 1 / 2)(24) n-1

41 Special Sequences Slide 121 / 145 Special Recursive Sequences Some recursive sequences not only rely on the preceding term, but on the two preceding terms. Find the first five terms of the sequence: a 1 = 4, a 2 = 7, and a n = a n-1 + a n = = =29 Special Sequences Slide 122 / 145 Special Recursive Sequences Some recursive sequences not only rely on the preceding term, but on the two preceding terms. Find the first five terms of the sequence: a 1 = 6, a 2 = 8, and a n = 2a n-1 + 3a n (8) + 3(6) = 34 2(34) + 3(8) = 92 2(92) + 3(34) = 286 Special Sequences Slide 123 / 145 Special Recursive Sequences Some recursive sequences not only rely on the preceding term, but on the two preceding terms. Find the first five terms of the sequence: a 1 = 10, a 2 = 6, and a n = 2a n-1 - a n (6) -10 = 2 2(2) - 6 = -2 2(-2) - 2= -6

42 Special Sequences Slide 124 / 145 Special Recursive Sequences Some recursive sequences not only rely on the preceding term, but on the two preceding terms. Find the first five terms of the sequence: a 1 = 1, a 2 = 1, and a n = a n-1 + a n = = =5 Special Sequences Slide 125 / 145 The sequence in the preceding example is called The Fibonacci Sequence 1, 1, 2, 3, 5, 8, 13, 21,... where the first 2 terms are 1's and any term there after is the sum of preceding two terms. This is as famous as a sequence can get and is worth remembering. Special Sequences Slide 126 / Find the first four terms of sequence: a 1 = 5, a 2 = 7, and a n = a 1 + a 2 A B C D 7, 5, 12, 19 5, 7, 35, 165 5, 7, 12, 19 5, 7, 13, 20

43 Special Sequences Slide 127 / Find the first four terms of sequence: a 1 = 4, a 2 = 12, and a n = 2a n-1 - a n-2 A B C D 4, 12, -4, -20 4, 12, 4, 12 4, 12, 20, 28 4, 12, 20, 36 Special Sequences Slide 128 / Find the first four terms of sequence: a 1 = 3, a 2 = 3, and a n = 3a n-1 + a n-2 A B C D 3, 3, 6, 9 3, 3, 12, 39 3, 3, 6, 21 3, 3, 12, 36 Slide 129 / 145 Binomial Theorem Return to Table of Contents

44 Binomial Theorem Slide 130 / 145 Look for a pattern when powering a binomial. Binomial Theorem Slide 131 / 145 One pattern comes from the coefficients This is Pascal's Triangle. Slide 132 / 145

45 Binomial Theorem Slide 133 / 145 Example: Find the coefficient of the 4 term of 6th power of a binomial. Binomial Theorem Slide 134 / Calculate Binomial Theorem Slide 135 / Calculate

Binomial Theorem 78 Calculate Slide 136 / 145 Binomial Theorem Slide 137 / 145 79 What is the binomial coefficient of the 4th term of

46 Binomial Theorem 78 Calculate Slide 136 / 145 Binomial Theorem Slide 137 / What is the binomial coefficient of the 4th term of x + y to the 8th power? Binomial Theorem Slide 138 / What is the binomial coefficient of the 2nd term of x + y to the 3rd power?

47 Binomial Theorem Slide 139 / 145 The second pattern of powering a binomial comes from the exponents. Click to see the Binomial Expansion. But what if instead of x + y we have 2x -3? Binomial Theorem Slide 140 / 145 The Binomial Theorem Binomial Theorem Slide 141 / 145 Example: Expand (2x -3) 4

48 Slide 142 / 145 Binomial Theorem Slide 143 / What is the coefficient of the 4th term (2x - 3) to the 5th power? Binomial Theorem Slide 144 / What is the coefficient of the 3rd term (2x - 3) to the 6th power?

49 Binomial Theorem Slide 145 / What is the coefficient of the 6th term (2x - 3) to the 7th power?

Chapter 8 Sequences, Series, and the Binomial Theorem

Chapter 8 Sequences, Series, and the Binomial Theorem Section 1 Section 2 Section 3 Section 4 Sequences and Series Arithmetic Sequences and Partial Sums Geometric Sequences and Series The Binomial Theorem