Slide 1 / 145 Slide 2 / 145 Pre-Calculus Sequences and Series 2015-03-24 www.njctl.org Slide 3 / 145 Table of Contents click on the topic to go to that section Arithmetic Sequences Arithmetic Series Geometric Sequences Geometric Series Infinite Geometric Series Special Sequences Binomial Theorem
Slide 4 / 145 Arithmetic Sequences Return to Table of Contents Slide 5 / 145 Slide 6 / 145
Slide 7 / 145 Arithmetic Sequence Teacher Teacher 1 Find the next term in the arithmetic sequence: 3, 9, 15, 21, . . . Slide 8 / 145 Arithmetic Sequence Teacher Teacher 2 Find the next term in the arithmetic sequence: -8, -4, 0, 4, . . . Slide 9 / 145 Arithmetic Sequence Teacher Teacher 3 Find the next term in the arithmetic sequence: 2.3, 4.5, 6.7, 8.9, . . .
Slide 10 / 145 Arithmetic Sequence Teacher Teacher 4 Find the value of d in the arithmetic sequence: 10, -2, -14, -26, . . . Slide 11 / 145 Arithmetic Sequence Teacher Teacher 5 Find the value of d in the arithmetic sequence: -8, 3, 14, 25, . . . Slide 12 / 145 Arithmetic Sequence Arithmetic Sequence 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.
Slide 13 / 145 Arithmetic Sequence Arithmetic Sequence Teacher Teacher 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 / 145 Arithmetic Sequence Teacher Teacher 6 Which sequence matches the description? A 4, 6, 8, 10 2, 6,10, 14 B 2, 8, 32, 128 C D 4, 8, 16, 32 Slide 15 / 145 Arithmetic Sequence Teacher Teacher 7 Which sequence matches the description? A -3, -7, -10, -14 -4, -7, -11, -13 B -3, -7, -11, -15 C -3, 1, 5, 9 D
Slide 16 / 145 Arithmetic Sequence Teacher Teacher 8 Which sequence matches the description? A 7, 10, 13, 16 4, 7, 10, 13 B C 1, 4, 7,10 3, 5, 7, 9 D Slide 17 / 145 Arithmetic Sequence Arithmetic Sequence 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. Slide 18 / 145 Arithmetic Sequence Arithmetic Sequence Teacher Teacher Consider: 3, 9, 15, 21, 27, 33, 39,. . . a 1 3 a 2 9 = 3+6 a 3 15 = 3+12 = 3+2(6) Do you see a pattern that relates the term number to its a 4 21 = 3+18 = 3+3(6) value? a 5 27 = 3+24 = 3+ 4(6) a 6 33 = 3+30 = 3+5(6) click a 7 39 = 3+36 = 3+6(6)
Slide 19 / 145 Arithmetic Sequence Teacher Teacher Arithmetic Sequence 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 Arithmetic Sequence Teacher Teacher Arithmetic Sequence 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 Arithmetic Sequence Arithmetic Sequence Teacher Teacher 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.
Slide 22 / 145 Arithmetic Sequence Teacher Teacher 9 Find a 11 when a 1 = 13 and d = 6. Slide 23 / 145 Arithmetic Sequence Teacher Teacher 10 Find a 17 when a 1 = 12 and d = -0.5 Slide 24 / 145 Arithmetic Sequence Teacher Teacher 11 Find a 17 for the sequence 2, 4.5, 7, 9.5, ...
Slide 25 / 145 Arithmetic Sequence 12 Find the common difference d when a 1 = 12 and a 13 = 6. Teacher Teacher Slide 26 / 145 Arithmetic Sequence Teacher Teacher 13 Find n such a 1 = 12 , a n = -20, and d = -2. Slide 27 / 145 Arithmetic Sequence Teacher Teacher 14 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?
Slide 28 / 145 Arithmetic Sequence Arithmetic Sequence Teacher Teacher Find the missing terms in the arithmetic sequence. 4, 6, 8, 10,___ 5, 10, ___, 20 ___, 12, 9, 6 6,___, 14 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 Slide 29 / 145 Arithmetic Sequence Teacher Teacher Arithmetic Sequence Find the missing terms 2, ___ , ___ , 23 4, ___ , ___ , -14 7, ___ , ___, ___, 39 -9, ___ , ___ , ___, ___ , -34 Slide 30 / 145 Arithmetic Sequence Teacher Teacher 15 Find the missing term: 4, ___ , -16 -20 A B -10 C -6 2 D
Slide 31 / 145 Arithmetic Sequence Teacher Teacher 16 Find the missing terms: -10, ___ , ___, 8 A -6, -2 -6, 2 B -5, 1 C D -4, 2 Slide 32 / 145 Arithmetic Sequence 17 Find the missing terms: 12, ___ , ___, 75 Teacher Teacher 27, 51 A B 33, 54 C 37, 51 D 34, 58 Slide 33 / 145 Arithmetic Sequence Teacher Teacher 18 Find d for the arithmetic sequence: 5, ___ , ___ , ___ , 21
Slide 34 / 145 Arithmetic Series Return to Table of Contents Slide 35 / 145 Arithmetic Series 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 = 4 + 7 + 10 + 13 = 34 S 6 = 4 + 7 + 10 + 13 + 16 + 19= 69 Slide 36 / 145 Arithmetic Series Teacher Teacher 19 Consider 3, 9, 15, 21, 27, 33, 39,... what is S 4 ?
Slide 37 / 145 Arithmetic Series Teacher Teacher 20 Consider 3, 9, 15, 21, 27, 33, 39,... what is S 5 ? Slide 38 / 145 Arithmetic Series Teacher Teacher 21 Consider 3, 9, 15, 21, 27, 33, 39,... what is S 7 ? Slide 39 / 145 Arithmetic Series 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 = 4 + 7 + 10 +13 + . . . + 292 + 295 +298 + 301 If we add the smallest and largest (4 + 301)= 305 If we add the next two (7 + 298) = 305 and continue (10 + 295) until they are all paired up (151+154). We now have 50 pairs of 305, so... S 100 = 4 + 7 + 10 +. . .+298 + 301 = 50(305) = 15250 Do you see a pattern?
Slide 40 / 145 Slide 41 / 145 Arithmetic Series Find S n for each arithmetic series. Teacher Teacher Example: a 1 = 7 and a 12 = -23 Example: a 1 = 6, n = 10, and d = 9 Slide 42 / 145 Arithmetic Series Find S n for each arithmetic series. Example: a 12 = 30 and d = -7 Teacher Teacher Example: a 1 = 2, a n = 32, and d = 5
Slide 43 / 145 Arithmetic Series 22 Find the S n for the arithmetic series described: Teacher Teacher a 1 = 19 and a 12 = 37. Slide 44 / 145 Arithmetic Series Teacher Teacher 23 Find the S n for the arithmetic series described: a 1 = 30 and a 17 = -45. Slide 45 / 145 Arithmetic Series Teacher Teacher 24 Find the S n for the arithmetic series described: a 1 = 20, n = 8, and d = 6.
Slide 46 / 145 Arithmetic Series Teacher Teacher 25 Find the S n for the arithmetic series described: a n = 20, n = 9, and d = -4. Slide 47 / 145 Arithmetic Series Teacher Teacher 26 Find the S n for the arithmetic series described: 17 + 20 + 23 + 26 + 29 + . . . + 50 Slide 48 / 145 Arithmetic Series 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.
Slide 49 / 145 Slide 50 / 145 Arithmetic Series Teacher Teacher 27 Evaluate Slide 51 / 145 Arithmetic Series Teacher Teacher 28 Evaluate
Slide 52 / 145 Arithmetic Series Teacher Teacher 29 Evaluate Slide 53 / 145 Arithmetic Series How would evaluate ? Teacher Teacher Slide 54 / 145 Arithmetic Series Teacher Teacher 30 Evaluate
Slide 55 / 145 Geometric Sequences Return to Table of Contents Slide 56 / 145 Slide 57 / 145 Geometric Sequence Teacher Teacher 31 Find the next term in geometric sequence: 6, -12, 24, -48, 96, . . .
Slide 58 / 145 Geometric Sequence Teacher Teacher 32 Find the next term in geometric sequence: 64, 16, 4, 1, . . . Slide 59 / 145 Geometric Sequence Teacher Teacher 33 Find the next term in geometric sequence: 6, 15, 37.5, 93.75, . . . Slide 60 / 145 Geometric Sequence Teacher Teacher 34 Is the following sequence a geomtric one? 48, 24, 12, 8, 4, 2, 1 Yes No
Slide 61 / 145 Geometric Sequence Geometric Sequence Geometric Sequences can be described by giving the Teacher Teacher 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 4) a 1 = 12 and r = 2 / 3 Slide 62 / 145 Geometric Sequence 35 Find the first four terms of the geometric sequence Teacher Teacher described: a 1 = 6 and r = 4. 6, 24, 96, 384 A 4, 24, 144, 864 B 6, 10, 14, 18 C D 4, 10, 16, 22 Slide 63 / 145 Geometric Sequence 36 Find the first four terms of the geometric sequence Teacher Teacher described: a 1 = 12 and r = - 1 / 2 . A 12, -6, 3, -.75 B 12, -6, 3, -1.5 6, -3, 1.5, -.75 C -6, 3, -1.5, .75 D
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