Arithmetic Sequences 1 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. 2 USING AND WRITING SEQUENCES DOMAIN: n an RANGE: 1 3 2 6 3 9 4 12 5 15 The domain gives the relative position of each term. 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. 3 Writing Terms of Sequences Write the first six terms of the sequence an = 2n + 3. SOLUTION a 1 = 2(1) + 3 = 5 1st term a 2 = 2(2) + 3 = 7 2nd term a 3 = 2(3) + 3 = 9 3rd term a 4 = 2(4) + 3 = 11 4th term a 5 = 2(5) + 3 = 13 5th term a 6 = 2(6) + 3 = 15 6th term 4 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 5 An introduction………… 1, 4, 7, 10, 13 d=3 9, 1, 7, 15 d = -8 6.2, 6.6, 7, 7.4 d = .4 , 3, 6 d = 3 ARITHMETIC 2, 4, 8, 16, 32 r =2 1 9, 3, 1, 1/ 3 r = 3 1 1, 1/ 4, 1/16, 1/ 64 r = 4 , 2.5, 6.25 r = 2.5 GEOMETRIC MULTIPLY ADD (by the same #) (by the same #) To get the next term To get the next term 6 Vocabulary of Sequences (Universal) a1 First term an nth term an-1 previous term an+1 next term n number of terms d common difference Finite VS. Infinite 7 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 8 Find the next 4 terms of –9, -2, 5, … 2 9 5 2 7 7 is referred to as d Next four terms…… 12, 19, 26, 33 9 Find the next four terms of 0, 7, 14, … Arithmetic Sequence, d = 7 21, 28, 35, 42 Find the next four terms of x, 2x, 3x, … Arithmetic Sequence, d = x 4x, 5x, 6x, 7x Find the next four terms of 5k, -k, -7k, … Arithmetic Sequence, d = -6k -13k, -19k, -25k, -31k 10 4, 10, 16, 22 a1 4 4 0(6) a2 4 6 4 1(6) a3 4 6 6 4 2(6) a4 4 6 6 6 4 3(6) The nth term of an arithmetic sequence is given by: an a1 (n 1)d The nth term in the sequence First term The term # The common difference Find the 10th term: a10 4 (10 1)6 4 54 58 11 Find the 14th term of the sequence: 4, 7, 10, 13,…… an a1 (n 1)d a14 4 (14 1)3 4 (13)3 43 12 In the arithmetic sequence 4,7,10,13,…, which term has a value of 301? an a1 (n 1)d 301 4 (n 1)3 301 4 3n 3 301 1 3n 100 n 300 3n 13 Given an arithmetic sequence with a15 38 and d 3, find a1. an a1 n 1 d 38 x 15 1 3 X = 80 14 Find d if a1 6 and a29 20 20 6 29 1 x 26 28x 13 x 14 15 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, 15 = a1 + (5 – 1)4 15 = a1 +16 a1 = –1 a1=? a10 = –1 + (10 – 1)4 = -1 + 36 a10 = 35 16 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 17 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 18 Explicit vs. Recursive Formulas 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 + d a1 = ___ an+1 = an + d a1 = ___ an = an-1 + 2 Ex: 4, 6, 8, 10… a1 = 4 19 Find the recursive formula for the following arithmetic sequence: 3, 8, 13, 18… an = an-1 + d a1 = 3 d = 5 an = an-1 + 5 a1 = 3 20 Using Recursive & Explicit Formulas an = an-1 + 6 a1 = 4 a2 = 4 + 6 = 10 a3 = 10 + 6 = 16 a4 = 16 + 6 = 22 a5 = 22 + 6 = 28 1. Create the 1st 5 terms: 4, 10, 16, 22, 28 2. Find the explicit formula: an = a1 + (n – 1)d an = 4 + (n – 1)6 an = 4 + 6n – 6 an = 6n – 2 21 Using Recursive & Explicit Formulas an = 7 – 2n 1. Create the a1 = 7 – 2(1) = 5 a2 = 7 – 2(2) = 3 a3 = 7 – 2(3) = 1 a4 = 7 – 2(4) = –1 a5 = 7 – 2(5) = –3 st 1 5 terms: 5, 3, 1, –1, –3 2. Find the recursive formula: an = an-1 – 2 a1 = 5 22 An arithmetic mean of two numbers, a and b, is simply their average. Use the formula and information given to find the common difference to create the sequence. Insert 3 arithmetic means between 8 & 16. 8, 10 , 12 , 14 ,16 Let 8 be the 1st term Let 16 be the 5th term Let 5 be N d is missing an a1 (n 1)d 16 8 (5 1)d d 2 23 Find two arithmetic means between –4 and 5 -4, ____, ____, 5 an a1 n 1 d 5 4 4 1 x x3 The two arithmetic means are –1 and 2, 24 since –4, -1, 2, 5 forms an arithmetic sequence Find 3 arithmetic means between 1 & 4 1, ____, ____, ____, 4 an a1 n 1 d 4 1 5 1 x 3 x 4 The 3 arithmetic means are since 7 10 13 1, , , 4 4 4 7 10 13 , , 4 4 4 ,4 forms a sequence 25 Geometric Sequences 26 Vocabulary of Sequences (Universal) a1 First term an nth term an-1 previous term an+1 next term n number of terms d common difference Finite VS. Infinite 27 Find the next 3 terms of 2, 3, 9/2, __, __, __ 3 – 2 vs. 9/2 – 3… not arithmetic a • Use n 1 to determine common ratio an 3 9/2 3 1.5 geometric r 2 3 2 28 How is the formula derived? 1st term: 2 The nth term of a geometric sequence is given by: a ar n1 3 2nd term : 2 3 n 1 2 3 3 9 3rd term : 2 2 2 2 4th term: 5th term: 6th term: 3 3 3 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 29 1 2 If a1 , r , find a9 . 2 3 9 1 n1 1 2 x 2 3 n 1 7 8 2 2 128 x 8 8 3 23 6561 a ar 30 2 Find a2 a 4 if a1 3 and r 3 4 8 2 -3, ____, 3 9 2 ____, ____ Since r ... 3 8 10 a 2 a 4 2 9 9 31 Find a9 of 2, 2, 2 2,... an a1r a= 2 r= 2 1 n=9 n1 2 2 2 9 1 x 2 x 8 x 16 2 32 If a5 32 2 and r 2, find a 2 ____, ____, ____,____,32 2 n1 an a1r 32 2 x 2 5 1 32 2 x 2 32 2 4x 8 2 x 4 a2 8 2 2 8 2 2 1 8 4 16 21 33 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: n-1 Ex: an = a1*r an = 4 * 3 n-1 34 Find the explicit formula for the following geometric sequence: 3, 6, 12, 24… an = a1*r n-1 a1 = 3 r =2 an = 3 *2 n-1 35 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 = an-1 (r) a1 = ___ a1 (r) = a2 a2 (r) = a3 a3 (r) = a4 an+1 = r(an) a1 = ___ Ex: –1, 4, –16, 64 … an = an-1 (–4) a1 = –1 36 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 37 Using Recursive & Explicit Formulas an = an-1 (3) a1 = –1 a2 = –1(3) = –3 a3 = –3(3) = –9 a4 = –9(3) = –27 a5 = –27(3) = –81 1. Create the 1st 5 terms: –1, –3, –9, –27, – 81 2. Find the explicit formula: n-1 an = a1 (r) n-1 an = –1(3) an = –3 n-1 38 Using Recursive & Explicit Formulas an = 2(4) n–1 1. Create the a1 = 2(4)1-1 = 2 a2 = 2(4)2-1 = 8 3-1 a3 = 2(4) = 32 4-1 a4 = 2(4) = 128 a5 = 2(4)5-1 = 512 st 1 5 terms: 2, 8, 32, 128, 512 2. Find the recursive formula: an = 4an-1 a1 = 2 39 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 an a1r n1 54 2 x 6 –18 -2, ____, ____, 54 The 2 geometric means are 6 and -18 27 x 3 x 3 40 41 *** Insert one geometric mean between ¼ and 4*** *** denotes trick question 1 ,____,4 4 an a1r n1 1 31 1 2 4 r 4 r 16 r 2 4 4 4 r 1 , 1, 4 4 1 , 1, 4 4 41 Series 42 Vocabulary of Sequences (Universal) a1 First term an nth term an-1 previous term an+1 next term n number of terms d common difference Finite VS. Infinite 43 USING SERIES When the terms of a sequence are added, the resulting expression is a series. A series can be finite or infinite. FINITE SEQUENCE INFINITE SEQUENCE 3, 6, 9, 12, 15 3, 6, 9, 12, 15, . . . FINITE SERIES INFINITE SERIES 3 + 6 + 9 + 12 + 15 ... 3 + 6 + 9 + 12 + 15 + . . . You can use summation notation to write a series. For example, for the finite series shown above, you can write 5 3 + 6 + 9 + 12 + 15 = ∑ 3i i=1 44 UPPER BOUND TERM NUMBER B a n SIGMA (SUM OF TERMS) n A # of Terms: B–A+1 NTH TERM SEQUENCE (EXPLICIT FORMULA) LOWER BOUND TERM NUMBER 45 j 2 18 4 j1 1 2 2 2 3 2 4 2 7 2a a4 44 2 4 2 5 2 6 2 7 46 An arithmetic series is a series associated with an arithmetic sequence. It can be infinite or finite. 47 1, 4, 7, 10, 13, …. 3, 7, 11, …, 51 Infinite Arithmetic (constantly getting larger or smaller) Finite Arithmetic No Sum n Sn a1 an 2 1, 2, 4, …, 64 1, 2, 4, 8, … 1 1 1 3,1, , , ... 3 9 27 48 Find the sum of the 1st 100 natural numbers. 1 + 2 + 3 + 4 + … + 100 a1 1 an 100 n 100 n S n a1 an 2 S100 100 (1 100) 2 5050 49 Find the sum of the 1st 14 terms of the series: 2 + 5 + 8 + 11 + 14 + 17 +… n S n a1 an 2 To find a14 , you need d 3 a1 2 n 14 a14 2 (14 1)d a14 = 2 + (14 - 1)(3) = 41 14 S14 2 a14 2 S14 = 14 2 41 301 2 50 13 Find the sum of the series (4n 5) 9 13 17 .... n 1 n Need 13th term: S n a1 an 2 4(13) + 5 = 57 a1 9 d 4 n 13 13 13 S13 9 57 S13 9 a13 2 2 13 (66) 429 2 51 Finding the Sum from Summation Notation n S n a1 an 2 4 j 2 n = 4 a1 = 3 a4 = 6 j1 3, 4, 5, 6 4 S 4 3 6 2(9) 18 2 7 2a a4 8, 10, 12, 14 n = (7 – 4) + 1 a4 = 8 a7 = 14 4 S 4 8 14 2(22) 44 2 52 19 4b 3 b 4 23 a4 =19 19, 23, 27, 31…79 a19 = 79 n = (19 - 4) + 1 = 16 16 S16 (19 79) 8(98) 784 2 2x 1 15, 17, 19, …47 x 7 a7 =15 a23 = 47 n = (23-7) + 1 = 17 17 S17 (15 47) 8.5(62) 527 2 53 An geometric series is a series associated with a geometric sequence. They can be infinite or finite. Finite and infinite have different formulas depending on the value of r. 54 1, 4, 7, 10, 13, …. Infinite Arithmetic (constantly getting larger or smaller) 3, 7, 11, …, 51 Finite Arithmetic No Sum n Sn a1 an 2 a1 (1 r ) Sn 1 r n 1, 2, 4, …, 64 1, 2, 4, 8, … Finite Geometric Infinite Geometric r < -1 OR r > 1 (constantly getting larger or smaller) “diverges” Infinite Geometric 1 1 1 3, 1, , , ... -1 < r < 1 3 9 27 “converges” No Sum a1 S 1 r 55 1 1 1 Find S7 of ... 2 4 8 Finite : Find sum of 1st 7 terms 1 1 1 1 8 4 n r a1 (1 r ) a1 Sn 1 1 2 2 r ? 1 r n7 2 4 1 1 7 (1 ( ) ) 127 2 2 S7 1 128 1 2 56 Sums of Infinite Series Made Finite (referred to as partial sums) Infinite Series Finding the Sum of Infinite Sequences “Converges” vs. “Diverges” 57 58 Find the sum, if possible: Geometric ~need to find r~ 1 1 1 2 4 r 1 1 2 2 S 1 1 1 2 1 1 1 1 ... 2 4 8 Is -1 < r < 1? Yes (Infinite Series - converges) a1 S 1 1 r 2 1 2 59 Find the sum, if possible: 2 2 8 16 2 ... 8 16 2 r 2 2 8 2 2 Is -1 < r < 1? No (Infinite series - Diverges) NO SUM 60 Find the sum, if possible: 4 2 2 9 3 r 2 1 3 3 S 2 4 8 1 ... 3 9 27 Is -1 < r < 1? Yes (Infinite Series – Converges) 1 1 3 2 3 a1 S 1 r 61 Find the sum, if possible: 2 4 8 ... 7 7 7 4 8 7 7 r 2 Is -1 < r < 1? No 2 4 (Infinite Series–Diverges) 7 7 NO SUM 62 Find the sum, if possible: 5 10 5 ... 2 5 5 1 2 r 10 5 2 10 S 1 1 2 -1 < r < 1 Yes (Infinite Series–Converges) 20 a1 S 1 r 63 Finding the Sum from Sigma Notation n 3 3 ... 3 3 6 6 6 6 5 5 5 b 0 5 23 3 r 5 so “converges” 0 1 2 a1 6 15 S 3 1 r 1 5 2x 1 27 1 28 1 29 1 ... 223 1 x 7 15, 17, 19, ...47 n 23 7 1 Sn a1 an 15 47 527 2 2 64 Rewrite using sigma notation: 3 + 6 + 9 + 12 1st term 4th term Arithmetic, d= 3 an a1 n 1 d an 3 n 1 3 an 3n Explicit formula 4 3n n=1 65 Rewrite using sigma notation: 16 + 8 + 4 + 2 + 1 1st term 5th term Geometric, r = ½ an a1r n1 1 an 16 2 Explicit formula n1 5 1 16 2 n=1 66 n 1 Rewrite the following using sigma notation: 3 9 27 ... Numerator is geometric, r = 3 5 10 15 Denominator is arithmetic d= 5 NUMERATOR: 3 9 27 ... an 3 3 n1 DENOMINATOR: 5 10 15 ... an 5 n 1 5 an 5n n1 SIGMA NOTATION: n1 3 3 5n 67