Arithmetic Sequences

1. Arithmetic Sequences

2. USING AND WRITING SEQUENCES The numbers in sequences are called terms. You can think of a sequence as a function whose domain is a set of consecutive integers. If a domain is not specified, it is understood that the domain starts with 1.

3. USING AND WRITING SEQUENCES n an 1 2 3 4 5 DOMAIN: The domain gives the relative positionof each term. The range gives the terms of the sequence. 3 6 9 12 15 RANGE: This is a finite sequence having the rule an= 3n, where anrepresents the nth term of the sequence.

4. Writing Terms of Sequences Write the first six terms of the sequence an = 2n + 3. SOLUTION a1= 2(1) + 3 = 5 1st term a2= 2(2) + 3 = 7 2nd term a3= 2(3) + 3 = 9 3rd term a4= 2(4) + 3 = 11 4th term a5= 2(5) + 3 = 13 5th term a6= 2(6) + 3 = 15 6th term

5. Writing Terms of Sequences Write the first six terms of the sequence f(n) = (–2)n – 1 . SOLUTION f(1) = (–2)1 – 1 = 1 1st term f(2) = (–2)2 – 1 = –2 2nd term f(3) = (–2)3 – 1 = 4 3rd term f(4) = (–2)4 – 1 = – 8 4th term f(5) = (–2)5 – 1 = 16 5th term f(6) = (–2)6 – 1 = – 32 6th term

6. An introduction………… r =2 d = 3 r = d = -8 r = d = .4 d = r = GEOMETRIC MULTIPLY (by the same #) To get the next term ARITHMETIC ADD (by the same #) To get the next term

7. an-1 previous term an+1 next term Vocabulary of Sequences (Universal) Finite VS. Infinite

8. Arithmetic Sequence: sequence whose consecutive terms have a common difference. Example:3, 5, 7, 9, 11, 13, ... The terms have a common difference of 2. (known as d) To find the common difference you use an+1 – an Example: Is the sequence arithmetic? If so, find d.–45, –30, –15, 0, 15, 30 d = 15

9. Find the next 4 terms of –9, -2, 5, … 7 is referred to as d Next four terms…… 12, 19, 26, 33

10. Find the next four terms of 0, 7, 14, … Find the next four terms of x, 2x, 3x, … Find the next four terms of 5k, -k, -7k, … Arithmetic Sequence, d = 7 21, 28, 35, 42 Arithmetic Sequence, d = x 4x, 5x, 6x, 7x Arithmetic Sequence, d = -6k -13k, -19k, -25k, -31k

11. The nth term in the sequence First term The term # The common difference 4, 10, 16, 22 The nth term of an arithmetic sequence is given by: Find the 10th term:

12. Examples: Find the 14th term of the sequence: 4, 7, 10, 13,……

13. Examples: In the arithmetic sequence 4,7,10,13,…, which term has a value of 301?

14. Given an arithmetic sequence with X = 80

16. Example: If the common difference is 4 and the fifth term is 15, what is the 10th term of an arithmetic sequence? an = a1 + (n – 1)d d = 4, a5 = 15, n = 5, a1=? 15 = a1 + (5 – 1)4 15 =a1 +16 a1 = –1 a10 = –1 + (10 – 1)4 = -1 + 36 a10 = 35

17. Explicit vs. Recursive Formulas Explicit Formula – used to find the nth term of the arithmetic sequence in which the common difference and 1st term are known. Ex: 4, 6, 8, 10… Use a1 and d in sequence formula: an = 4 + (n – 1)2 an = 2n + 2

18. Find the explicit formula for the following arithmetic sequence: 3, 8, 13, 18… an = a1 + (n – 1)d a1 = 3 d = 5 n = ? an = 3 + (n – 1)5 an = 3 + 5n – 5 an= -2 + 5n OR an = 5n – 2

19. Explicit vs. Recursive Formulas an = an-1 + d a1 = ___ an+1 = an+ d a1 = ___ Recursive Formula – (includes a1) used to find the next term of the sequence by adding the common difference to the previous term. an = an-1 + 2 a1 = 4 Ex: 4, 6, 8, 10…

20. Find the recursive formula for the following arithmetic sequence: 3, 8, 13, 18… an = an-1 + da1= 3d = 5 an = an-1 +5 a1 = 3

21. Using Recursive & Explicit Formulas 1. Create the 1st 5 terms: an = an-1 +6 a1 = 4 4, 10, 16, 22, 28 2. Find the explicit formula: a2 = 4 + 6 = 10 an = a1 + (n – 1)d a3 = 10 + 6 = 16 an = 4 + (n – 1)6 a4 = 16 + 6 = 22 an = 4 + 6n – 6 an = 6n – 2 a5 = 22 + 6 = 28

22. Using Recursive & Explicit Formulas an = 7 – 2n 1. Create the 1st 5 terms: 5, 3, 1, –1, –3 a1 = 7 – 2(1) = 5 a2= 7 – 2(2) = 3 2. Find the recursive formula: a3= 7 – 2(3) = 1 a4= 7 – 2(4) = –1 an = an-1 – 2 a1 = 5 a5= 7 – 2(5) = –3

23. Examples: Insert 3 arithmetic means between 8 & 16. Let 8 be the 1st term Let 16 be the 5th term Let 5 be N d is missing 14 An arithmetic meanof two numbers, a and b, is simply their average. Use the formula and information given to find the common difference to create the sequence. 10 12

24. The two arithmetic means are –1 and 2, since –4, -1, 2, 5 forms an arithmetic sequence Find two arithmetic means between –4 and 5 -4, ____, ____, 5

25. The 3 arithmetic means are since 1, ,4 forms a sequence Find 3 arithmetic means between 1 & 4 1, ____, ____, ____, 4

26. Geometric Sequences

27. an-1 previous term an+1 next term Vocabulary of Sequences (Universal) Finite VS. Infinite

28. Use to determine common ratio Find the next 3 terms of 2, 3, 9/2, __, __, __ 3 – 2 vs. 9/2 – 3… not arithmetic

29. The nth term of a geometric sequence is given by: How is the formula derived? 1st term: 2 5th term: 6th term: 4th term:

31. -3, ____, ____, ____

32. r = a1= n = 9

33. 8 2 x = ( )2-1 = - a 8 2 2 2

34. Explicit vs. Recursive Formulas Explicit Formula – used to find the nth term of the geometric sequence in which the common ratio and 1st term are known. Ex: 4, 12, 36, 108… Use a1 and r in sequence formula: Ex: an = a1*rn-1an = 4 * 3n-1

35. Find the explicit formula for the following geometric sequence: 3, 6, 12, 24… an = a1*rn-1a1 = 3 r =2 an =3*2n-1

36. Explicit vs. Recursive Formulas Recursive Formula (includes a1) – used to find the next term of the sequence by multiplying the common ratio to the previous term. an+1 = r(an) a1 = ___ an = an-1(r) a1 = ___ Ex: –1, 4, –16, 64 … a1 (r) = a2 an = an-1(–4) a1 = –1 a2 (r) = a3 a3 (r) = a4

37. Find the recursive formula for the following geometric sequence: 3, 6, 12, 24… an = an-1 * r a1 = 3 r = 2 an = an-1 * 2 a1 = 3

38. Using Recursive & Explicit Formulas 1. Create the 1st 5 terms: an = an-1 (3) a1 = –1 –1, –3, –9, –27, – 81 2. Find the explicit formula: a2 = –1(3) = –3 an = a1 (r)n-1 a3 = –3(3) = –9 an = –1(3)n-1 a4 = –9(3) = –27 an = –3n-1 a5 = –27(3) = –81

39. Using Recursive & Explicit Formulas an = 2(4)n – 1 1. Create the 1st 5 terms: 2, 8, 32, 128, 512 a1 = 2(4)1-1 = 2 a2= 2(4)2-1 = 8 2. Find the recursive formula: a3= 2(4)3-1 = 32 a4= 2(4)4-1 = 128 an = 4an-1 a1 = 2 a5= 2(4)5-1 = 512

40. The 2 geometric means are 6 and -18 A geometric mean(s)of numbers are the terms between any 2 nonsuccessive terms of a geometric sequence. Use the terms given to find the common ratio and find the missing terms called the geometric means. Ex: Find two geometric means between –2 and 54 6 –18 -2, ____, ____, 54

41. *** Insert one geometric mean between ¼ and 4*** *** denotes trick question

42. Series

43. an-1 previous term an+1 next term Vocabulary of Sequences (Universal) Finite VS. Infinite

44. FINITE SEQUENCE INFINITE SEQUENCE 3, 6, 9, 12, 15 3, 6, 9, 12, 15, . . . INFINITE SERIES FINITE SERIES 3 + 6 + 9 + 12 + 15 + . . . 3 + 6 + 9 + 12 + 15 5 3 + 6 + 9 + 12 + 15 = ∑ 3i i = 1 USING SERIES When the terms of a sequence are added, the resulting expression is a series. A series can be finite or infinite. . . . You can use summation notation to write a series. For example, for the finite series shown above, you can write

45. UPPER BOUND TERM NUMBER SIGMA (SUM OF TERMS) NTH TERM SEQUENCE (EXPLICIT FORMULA) LOWER BOUND TERM NUMBER # of Terms: B – A + 1

47. An arithmetic seriesis a series associated with an arithmetic sequence. It can be infinite or finite. Definition:

48. Infinite Arithmetic (constantly getting larger or smaller) 1, 4, 7, 10, 13, …. No Sum 3, 7, 11, …, 51 Finite Arithmetic 1, 2, 4, …, 64 1, 2, 4, 8, …

49. Examples: Find the sum of the 1st 100 natural numbers. 1 + 2 + 3 + 4 + … + 100

50. Find the sum of the 1st 14 terms of the series: 2 + 5 + 8 + 11 + 14 + 17 +… Examples: S14= To find a14 , you need a14 = 2 + (14 - 1)(3) = 41