Arithmetic Sequences and Series-Boger.ppt
- 1. Arithmetic Sequences and Series
- 2. An introduction………… 1, 4, 7,10,13 9,1, 7, 15 6.2, 6.6, 7, 7.4 , 3, 6 Arithmetic Sequences ADD To get next term 2, 4, 8,16, 32 9, 3,1, 1/3 1,1/ 4,1/16,1/ 64 , 2.5 , 6.25 Geometric Sequences MULTIPLY To get next term Arithmetic Series Sum of Terms 35 12 27.2 3 9 Geometric Series Sum of Terms 62 20/3 85/ 64 9.75
- 3. 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.
- 4. The domain gives the relative position of each term. 1 2 3 4 5 DOMAIN: 3 6 9 12 15 RANGE: The range gives the terms of the sequence. This is a finite sequence having the rule an = 3n, where an represents the nth term of the sequence. USING AND WRITING SEQUENCES n an
- 5. Writing Terms of Sequences Write the first five terms of the sequence an = 2n + 3. SOLUTION a1 = 2(1) + 3 = 5 1st term 2nd term 3rd term 4th term a2 = 2(2) + 3 = 7 a3 = 2(3) + 3 = 9 a4 = 2(4) + 3 = 11 a5 = 2(5) + 3 = 13 5th term
- 6. Writing Terms of Sequences SOLUTION 3rd term Write the first five terms of the sequence . 1 n n an 2 1 1 ) 1 ( ) 1 ( 1 a 1st term 3 2 1 ) 2 ( ) 2 ( 2 a 2nd term 4 3 1 ) 3 ( ) 3 ( 3 a 5 4 1 ) 4 ( ) 4 ( 4 a 6 5 1 ) 5 ( ) 5 ( 5 a 4th term 5th term
- 7. Writing Terms of Sequences Write the first five terms of the sequence an = n! - 2. SOLUTION a1 = (1)!-2 = -1 1st term 2nd term 3rd term 4th term a2 = (2)! - 2 = 0 a3 = (3)! - 2 = 4 a4 = (4)! - 2 = 22 a5 = (5)! - 2 = 118 5th term
- 8. Writing Terms of Sequences Write the first five terms of the sequence f(n) = (–2)n – 1 . SOLUTION f(1) = (–2) 1 – 1 = 1 1st term 2nd term 3rd term 4th term f(2) = (–2) 2 – 1 = –2 f(3) = (–2) 3 – 1 = 4 f(4) = (–2) 4 – 1 = – 8 f(5) = (–2) 5 – 1 = 16 5th term
- 9. Arithmetic Sequences and Series 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. The common difference is the number d. To find the common difference you use an+1 – an Example: Is the sequence arithmetic? –45, –30, –15, 0, 15, 30 Yes, the common difference is 15
- 10. Find the next four terms of –9, -2, 5, … Arithmetic Sequence 7 is referred to as the common difference (d) Common Difference (d) – what we ADD to get next term Next four terms……12, 19, 26, 33 -2 - -9 = 7 and 5 - -2 = 7
- 11. Find the next four terms of 0, 7, 14, … Arithmetic Sequence, d = 7 21, 28, 35, 42 Find the next four terms of -3x, -2x, -x, … Arithmetic Sequence, d = x 0, x, 2x, 3x Find the next four terms of 5k, -k, -7k, … Arithmetic Sequence, d = -6k -13k, -19k, -25k, -31k
- 12. How do you find any term in this sequence? To find any term in an arithmetic sequence, use the formula an = a1 + (n – 1)d where d is the common difference.
- 13. Vocabulary of Sequences (Universal) 1 a First term n a nth term n S sum of n terms n number of terms d common difference n 1 n 1 n nth term of arithmetic sequence sum of n terms of arithmetic sequen a a n 1 d n S a a 2 ce
- 14. Find the 14th term of the arithmetic sequence 4, 7, 10, 13,…… 1 ( 1) n a a n d 14 a (14 1) 4 3 4 (13)3 4 39 43
- 15. 16 1 Find a if a 1.5 and d 0.5 Try this one: 1 a First term n a nth term n S sum of n terms n number of terms d common difference 1.5 16 a16 NA 0.5 n 1 a a n 1 d a16 = 1.5 + (16 - 1)(0.5) a16 = 1.5 + (15)(0.5) a16 = 1.5+7.5 a16 = 9
- 16. Months Cost ($) 1 75,000 2 90,000 3 105,000 4 120,000 The table shows typical costs for a construction company to rent a crane for one, two, three, or four months. Assuming that the arithmetic sequence continues, how much would it cost to rent the crane for twelve months? 75,000 12 a12 NA 15,000 1 a First term n a nth term n S sum of n terms n number of terms d common difference n 1 a a n 1 d a12 = 75,000 + (12 - 1)(15,000) a12 = 75,000 + (11)(15,000) a12 = 75,000+165,000 a12 = $240,000
- 17. 1 ( 1) n a a n d In the arithmetic sequence 4,7,10,13,…, which term has a value of 301? 301 4 ( 1)3 n 301 4 3 3 n 301 1 3n 300 3n 100 n
- 18. n 1 Findnif a 633, a 9, and d 24 1 a First term n a nth term n S sum of n terms n number of terms d common difference 9 n 633 NA 24 n 1 a a n 1 d Try this one: 633 = 9 + (n - 1)(24) 633 = 9 + 24n - 24 633 = 24n – 15 648 = 24n n = 27
- 19. Given an arithmetic sequence with 15 1 a 38 and d 3, find a . 1 a First term n a nth term n S sum of n terms n number of terms d common difference a1 15 38 NA -3 n 1 a a n 1 d 38 = a1 + (15 - 1)(-3) 38 = a1 + (14)(-3) 38 = a1 - 42 a1 = 80
- 20. 1 29 Find d if a 6 and a 20 -6 29 20 NA d 1 a First term n a nth term n S sum of n terms n number of terms d common difference n 1 a a n 1 d 20 = -6 + (29 - 1)(d) 20 = -6 + (28)(d) 26 = 28d 14 13 d
- 21. Write an equation for the nth term of the arithmetic sequence 8, 17, 26, 35, … 1 a First term d common difference 8 9 an = 8 + (n - 1)(9) an = 8 + 9n - 9 an = 9n - 1 n 1 a a n 1 d
- 22. Arithmetic Mean: The terms between any two nonconsecutive terms of an arithmetic sequence. Ex. 19, 30, 41, 52, 63, 74, 85, 96 41, 52, 63 are the Arithmetic Mean between 30 and 74
- 23. Find two arithmetic means between –4 and 5 -4, ____, ____, 5 1 a First term n a nth term n S sum of n terms n number of terms d common difference -4 4 5 NA d n 1 a a n 1 d The two arithmetic means are –1 and 2, since –4, -1, 2, 5 forms an arithmetic sequence 5 = -4 + (4 - 1)(d) 5 = -4 + (3)(d) 9 = (3)(d) d = 3
- 24. Find three arithmetic means between 21 and 45 21, ____, ____, ____, 45 1 a First term n a nth term n S sum of n terms n number of terms d common difference 21 5 45 NA d n 1 a a n 1 d The three arithmetic means are 27, 33, and 39 since 21, 27, 33, 39, 45 forms an arithmetic sequence 45 = 21 + (5 - 1)(d) 45 = 21 + (4)(d) 24 = (4)(d) d = 6
- 25. Arithmetic Series: An indicated sum of terms in an arithmetic sequence. Example: Arithmetic Sequence 3, 5, 7, 9, 11, 13 VS Arithmetic Series 3 + 5 + 7 + 9 + 11 + 13
- 26. Vocabulary of Sequences (Universal) 1 a First term n a nth term n S sum of n terms n number of terms d common difference Recall
- 27. 1 a First term n a nth term n S sum of n terms n number of terms d common difference -19 63 ?? Sn 6 n 1 a a n 1 d 353 n 1 n n S a a 2 63 63 3 3 S 2 19 5 63 1 1 S 052 Find the sum of the first 63 terms of the arithmetic sequence -19, -13, -7,… n 1 n n S a a 2 a63 = 353
- 28. Find the first 3 terms for an arithmetic series in which a1 = 9, an = 105, Sn =741. 1 a First term n a nth term n S sum of n terms n number of terms d common difference 9 ?? 105 741 ?? n 1 a a n 1 d n 1 n n S a a 2 13 n 1 a a n 1 d 9, 17, 25
- 29. A radio station considered giving away $4000 every day in the month of August for a total of $124,000. Instead, they decided to increase the amount given away every day while still giving away the same total amount. If they want to increase the amount by $100 each day, how much should they give away the first day? 1 a First term n a nth term n S sum of n terms n number of terms d common difference a1 31 days ?? $124,000 $100/day n 1 n n S a a 2 n 1 a a n 1 d n 1 n n S a a 2
- 30. Sigma Notation ( ) Used to express a series or its sum in abbreviated form.
- 31. B n n A a UPPER LIMIT (NUMBER) LOWER LIMIT (NUMBER) SIGMA (SUM OF TERMS) NTH TERM (SEQUENCE) INDEX
- 32. j 4 1 j 2 2 1 2 2 3 2 2 4 18 7 4 a 2a 4 2 2 5 2 6 7 2 44 If the sequence is arithmetic (has a common difference) you can use the Sn formula j 4 1 j 2 n 1 n n S a a 2 1 a First term n a nth term n S sum of n terms n number of terms d common difference 1+2=3 4 4+2=6 ?? NA
- 33. Is the sequence arithmetic? 10 + 17 + 26 + 37 No, there is no common difference Thus you cannot use the Sn formula. = 90 = 2.71828
- 34. Rewrite using sigma notation: 3 + 6 + 9 + 12 Arithmetic, d= 3 n 1 a a n 1 d n a 3 n 1 3 n a 3n 4 1 n 3n