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New Jersey Center for Teaching and Learning Progressive Mathematics Initiative.

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  1. New Jersey Center for Teaching and Learning Progressive Mathematics Initiative This material is made freely available at www.njctl.org 
and is intended for the non-commercial use of 
students and teachers. These materials may not be 
used for any commercial purpose without the written 
permission of the owners. NJCTL maintains its 
website for the convenience of teachers who wish to 
make their work available to other teachers, 
participate in a virtual professional learning 
community, and/or provide access to course 
materials to parents, students and others. Click to go to website: www.njctl.org

  2. Algebra II Sequences and Series 2013-09-16 www.njctl.org

  3. Table of Contents Click on the topic to go to that section Arithmetic Sequences Geometric Sequences Geometric Series Fibonacci and Other Special Sequences Sequences as functions

  4. Arithmetic 
Sequences Return to 
Table of 
Contents

  5. Goals and Objectives Students will be able to understand how the common difference leads to the next term of an arithmetic sequence, the explicit form for an Arithmetic sequence, and how to use the explicit formula to find missing data. Why Do We Need This? Arithmetic sequences are used to model patterns and form predictions for events based on these patterns, such as in loan payments, sales, and revenue.

  6. Vocabulary An Arithmetic sequence is the set of numbers found by adding the same value to get from one term to the next. Example: 1, 3, 5, 7,... 10, 20, 30,... 10, 5, 0, -5,..

  7. Vocabulary The common difference for an arithmetic sequence is the value being added between terms, and is represented by the variable d. Example: 1, 3, 5, 7,... d=2 10, 20, 30,... d=10 10, 5, 0, -5,.. d=-5

  8. Notation As we study sequences we need a way of naming the terms. a1 to represent the first term, a2 to represent the second term, a3 to represent the third term, and so on in this manner. If we were talking about the 8th term we would use a8. When we want to talk about general term call it the nth term and use an.

  9. Finding the Common Difference 1. Find two subsequent terms such as a1 and a2 2. Subtract a2 - a1 a2=10 a1=4 d=10 - 4 = 6 Find d: 4, 10, 16, ... Solution

  10. Find the common difference: 1, 4, 7, 10, . . . 5, 11, 17, 23, . . . 9, 5, 1, -3, . . . d=3 d=6 d= -4 d= 2 1/2 Solutions

  11. NOTE: You can find the common difference using ANY set of consecutive terms For the sequence 10, 4, -2, -8, ... Find the common difference using a1 and a2: Find the common difference using a3 and a4: What do you notice?

  12. To find the next term: 1. Find the common difference 2. Add the common difference to the last term of the sequence 3. Continue adding for the specified number of terms d=9-5=4 a5=13+4=17 a6=17+4=21 a7=21+4=25 Example: Find the next three terms 1, 5, 9, 13, ... Solution

  13. Find the next three terms: 1, 4, 7, 10, . . . 5, 11, 17, 23, . . . 9, 5, 1, -3, . . . 13, 16, 19 29, 35, 41 -7, -11, -15

  14. 1 Find the next term in the arithmetic sequence: 3, 9, 15, 21, . . . 27 Solution

  15. 2 Find the next term in the arithmetic sequence: -8, -4, 0, 4, . . . 8 Solution

  16. 3 Find the next term in the arithmetic sequence: 2.3, 4.5, 6.7, 8.9, . . . 11.1 Solution

  17. 4 Find the value of d in the arithmetic sequence: 10, -2, -14, -26, . . . d=-12 Solution

  18. 5 Find the value of d in the arithmetic sequence: -8, 3, 14, 25, . . . d=11 Solution

  19. Write the first four terms of the arithmetic sequence 
that is described. 1. Add d to a1 2. Continue to add d to each subsequent terms a1=3 a2=3+7=10 a3=10+7=17 a4=17+7=24 Example: Write the first four terms of the sequence: a1=3, d= 7 Solution

  20. Find the first three terms for the arithmetic sequence described: a1 = 4; d = 6 a1 = 3; d = -3 a1 = 0.5; d = 2.3 a2 = 7; d = 5 1. 4,10, 16, ... 2. 3, 0, -3, ... 3. .5, 3.8, 6.1, ... 4. 7, 12, 17, ... Solution

  21. 6 Which sequence matches the description? A 4, 6, 8, 10 B B 2, 6,10, 14 Solution C 2, 8, 32, 128 D 4, 8, 16, 32

  22. 7 Which sequence matches the description? C A -3, -7, -10, -14 Solution B -4, -7, -10, -13 C -3, -7, -11, -15 D -3, 1, 5, 9

  23. 8 Which sequence matches the description? A 7, 10, 13, 16 A Solution B 4, 7, 10, 13 C 1, 4, 7,10 D 3, 5, 7, 9

  24. Recursive Formula To write the recursive formula for an arithmetic sequence: 1. Find a1 2. Find d 3. Write the recursive formula:

  25. Example: Write the recursive formula for 1, 7, 13, ... a1=1 d=7-1=6 Solution

  26. Write the recursive formula for the following sequences: a1 = 3; d = -3 a1 = 0.5; d = 2.3 Solution 1, 4, 7, 10, . . . 5, 11, 17, 23, . . .

  27. 9 Which sequence is described by the recursive formula? A -2, -8, -16, ... B -2, 2, 6, ... C 2, 6, 10, ... D 4, 2, 0, ...

  28. 10 A recursive formula is called recursive because it uses the previous term. True False

  29. 11 Which sequence matches the recursive formula? A -2.5, 0, 2.5, ... B -5, -7.5, -9, ... C -5, -2.5, 0, ... D -5, -12.5, -31.25, ...

  30. Arithmetic Sequence To find a specific term,say the 5th or a5, you could write out all of the terms. But what about the 100th term(or a100)? We need to find a formula to get there directly 
without writing out the whole list. DISCUSS: Does a recursive formula help us solve this problem?

  31. Arithmetic Sequence Consider: 3, 9, 15, 21, 27, 33, 39,. . . Do you see a pattern that 
relates the term number to its 
value?

  32. This formula is called the explicit formula. It is called explicit because it does not depend on the previous term The explicit formula for an arithmetic sequence is:

  33. To find the explicit formula: 1. Find a1 2. Find d 3. Plug a1 and d into 4. Simplify a1=4 d= -1-4 = -5 an= 4+(n-1)-5 an=4-5n+5 an=9-5n Example: Write the explicit formula for 4, -1, -6, ... Solution

  34. Write the explicit formula for the sequences: 1) 3, 9, 15, ... 2) -4, -2.5, -1, ... 3) 2, 0, -2, ... 1. an = 3+(n-1)6 = 3+6n-6 an=6n-3 2. an= -4+(n-1)2.5 = -4+2.5n-2.5 an=2.5n-6.5 3. an=2+(n-1)(-2)=2-2n+2 an=4-2n Solution

  35. 12 The explicit formula for an arithmetic sequence requires knowledge of the previous term True False False Solution

  36. 13 Find the explicit formula for 7, 3.5, 0, ... A B B Solution C D

  37. 14 Write the explicit formula for -2, 2, 6, .... A B D C Solution D

  38. 15 Which sequence is described by: A 7, 9, 11, ... A Solution B 5, 7, 9, ... C 5, 3, 1, ... D 7, 5, 3, ...

  39. 16 Find the explicit formula for -2.5, 3, 8.5, ... A B D Solution C D

  40. 17 What is the initial term for the sequence described by: -7.5 Solution

  41. Finding a Specified Term 1. Find the explicit formula for the sequence. 2. Plug the number of the desired term in for n 3. Evaluate Example: Find the 31st term of the sequence described by n=31 a31=3+2(31) a31=65 Solution

  42. Example Find the 21st term of the arithmetic sequence with a1 = 4 and d = 3. an = a1 +(n-1)d a21 = 4 + (21 - 1)3  a21 = 4 + (20)3  a21 = 4 + 60  a21 = 64 Solution

  43. Example Find the 12th term of the arithmetic sequence with a1 = 6 and d = -5. an = a1 +(n-1)d  a12 = 6 + (12 - 1)(-5)  a12 = 6 + (11)(-5)  a12 = 6 + -55  a12 = -49 Solution

  44. Finding the Initial Term or Common Difference 1. Plug the given information into an=a1+(n-1)d 2. Solve for a1, d, or n Example: Find a1 for the sequence described by a13=16 and d=-4 an = a1 +(n-1)d  16 = a1+ (13 - 1)(-4)  16 = a1 + (12)(-4)  16 = a1 + -48 a1 = 64 Solution

  45. Example Find the 1st term of the arithmetic sequence with a15 = 30 and d = 7. an = a1 +(n -1)d 30 = a1 + (15 - 1)7 30 = a1 + (14)7 30 = a1 + 98 -58 = a1 Solution

  46. Example Find the 1st term of the arithmetic sequence with a17 = 4 and d = -2. an = a1 +(n-1)d 4 = a1 + (17- 1)(-2) 4 = a1 + (16)(-2) 4 = a1 + -32 36 = a1 Solution

  47. Example Find d of the arithmetic sequence with a15 = 45 and a1=3. an = a1 +(n -1)d 45 = 3 + (15 - 1)d 45 = 3 + (14)d 42 = 14d 3 = d Solution

  48. Example Find the term number n of the arithmetic sequence with an = 6, a1=-34 and d = 4. an = a1 +(n-1)d 6 = -34 + (n- 1)(4) 6 = -34 + 4n -4 6 = 4n + -38 44 = 4n 11 = n Solution

  49. 18 Find a11 when a1 = 13 and d = 6. an = a1 +(n-1)d a11= 13 + (11- 1)(6) a11 = 13 + (10)(6) a11 = 13+60 a11 = 73 Solution

  50. 19 Find a17 when a1 = 12 and d = -0.5 an = a1 +(n-1)d a17= 12 + (17- 1)(-0.5) a17 = 12 + (16)(-0.5) a17 = 12+(-8) a17 = 4 Solution

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