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Sequences Sequence There are 2 types of Sequences Arithmetic: You add a common difference each time. Geometric: You multiply a common ratio each time. Arithmetic Sequences Example: • {2, 5, 8, 11, 14, ...} Add 3 each time • {0, 4, 8, 12, 16, ...} Add 4 each time • {2, -1, -4, -7, -10, ...} Add –3 each time Arithmetic Sequences • Find the 7th term of the sequence: 2,5,8,… Determine the pattern: Add 3 (known as the common difference) Write the new sequence: 2,5,8,11,14,17,20 So the 7th number is 20 Arithmetic Sequences • When you want to find a large sequence, this process is long and there is great room for error. • To find the 20th, 45th, etc. term use the following formula: an = a1 + (n - 1)d Arithmetic Sequences an = a1 + (n - 1)d Where: a1 is the first number in the sequence n is the number of the term you are looking for d is the common difference an is the value of the term you are looking for Arithmetic Sequences • Find the 15th term of the sequence: 34, 23, 12,… Using the formula an = a1 + (n - 1)d, a1 = 34 d = -11 n = 15 an = 34 + (n-1)(-11) = -11n + 45 a15 = -11(15) + 45 a15 = -120 Arithmetic Sequences Melanie is starting to train for a swim meet. She begins by swimming 5 laps per day for a week. Each week she plans to increase her number of daily laps by 2. How many laps per day will she swim during the 15th week of training? Arithmetic Sequences • What do you know? an = a1 + (n - 1)d a1 = 5 d= 2 n= 15 t15 = ? Arithmetic Sequences • tn = t1 + (n - 1)d • tn = 5 + (n - 1)2 • tn = 2n + 3 • t15 = 2(15) + 3 • t15 = 33 During the 15th week she will swim 33 laps per day. Geometric Sequences • In geometric sequences, you multiply by a common ratio each time. • 1, 2, 4, 8, 16, ... multiply by 2 27, 9, 3, 1, 1/3, ... Divide by 3 which means multiply by 1/3 • Geometric Sequences • Find the 8th term of the sequence: 2,6,18,… Determine the pattern: Multiply by 3 (known as the common ratio) Write the new sequence: 2,6,18,54,162,486,1458,4374 So the 8th term is 4374. Geometric Sequences • Again, use a formula to find large numbers. • an = a1 • (r)n-1 Geometric Sequences • Find the 10th term of the sequence : 4,8,16,… an = a1 • (r)n-1 • a1 = 4 • r=2 • n = 10 Geometric Sequences an = a1 • (r)n-1 a10 = 4 • (2)10-1 a10 = 4 • (2)9 a10 = 4 • 512 a10 = 2048 Geometric Sequences • Find the ninth term of a sequence if a3 = 63 and r = -3 a1= ? n= 9 r = -3 a9 = ? There are 2 unknowns so you must… Geometric Sequences • First find t1. • Use the sequences formula substituting t3 in for tn. a3 = 63 • a3 = a1 • (-3)3-1 • 63 = a1 • (-3)2 • 63= a1 • 9 • 7 = a1 Geometric Sequences • Now that you know t1, substitute again to find tn. an = a1 • (r)n-1 a9 = 7 • (-3)9-1 a9 = 7 • (-3)8 a9 = 7 • 6561 a9 = 45927 Sequence A sequence is a set of numbers in a specific order Infinite sequence Finite sequence a1 , a2 , a3 , a4 ,..., an ,... a1 , a2 , a3 , a4 ,..., an Sequences – sets of numbers Notation: an represents the formula for finding terms n term number a4 is the notation for the 4th term a32 is the notation for the 32nd term Examples: If an 2n 3, find the first 5 terms. If an 3n 1, find the 20th term. . Ex 1 Find the first four terms of the sequence an 3n 2 a1 3(1) 2 1 First term a2 4 Second term a3 7 Third term a4 10 Fourth term Ex. 2 Find the first four terms of the sequence (1) n an 2n 1 Writing Rules for Sequences We can calculate as many terms as we want as long as we know the rule or equation for an. Example: 3, 5, 7, 9, ___ , ___,……. _____ . an = 2n + 1 Writing Rules for Sequences Try these!!! 3, 6, 9, 12, ___ , ___,……. _____ . 1/1, 1/3, 1/5, 1/7, ___ , ___,……. _____ . an = 3n, an = 1/(2n-1)