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Notes Lesson 3-5 Arithmetic Sequences as Linear Functions Sequence – a set of numbers in a particular order Terms of the sequence – the numbers in a sequence Arithmetic sequence – is a sequence where the difference between the successive terms is constant Common difference – the difference between the terms, d Key Concept: an arithmetic sequence is a numerical pattern that increases or decreases at a constant rate called the common difference. ex. 3, 5, 7, 9, 11… 33, 29, 25, 21, 17 … the three dots used with sequences are called an ellipsis and it indicates that there are more terms that are not listed. Determine whether each sequence is an arithmetic sequence: a. -4, -2, 0, 2, … b. 1 5 3 13 , , , 2 8 4 16 ,… Find the next three terms of the sequence: 15, 9, 3, -3, … a. Step one: find the common difference (15 – 9 = 6, 9 – 3 = 6) so d = -6 b. Step two: add d to the last term of the sequence to get the next term (-3 + -6 = -9, -9 + -6 = -15, … c. Repeat. Find the next four terms of the arithmetic sequence: 9.5, 11.0, 12.5, 14.0… Each term in an arithmetic sequence can be expressed in terms of the first term, a₁, and the common difference, d. Term Symbol In terms of a₁ and d Numbers First term Second term Third term Fourth term a₁ a₂ a₃ a₄ a₁ a1 + d a₁ + 2d a₁ + 3d 8 8 + (3)= 11 8 + 2(3) = 14 8 + 3(3) = 17 nth term an a₁ + (n-1)d 8 + (n-1)(3) Example: Find the nth term a. Write an equation for the nth term of the arithmetic sequence -12, -8, -4, 0, … Step 1: find the common difference -12 -8 -4 0 +4 +4 +4 Step 2: Write an equation: an = a1 + (n-1)d = -12 + (n-1) 4 = -12 + 4n – 4 = 4n – 16 b. Find the 9th term of the sequence Substitute 9 for n in the formula an = 4n – 16 a9 = 4(9) – 16 a9 = 36 – 16 a9 = 20 c. Graph the first 5 terms of the sequence n 4n – 16 1 4(1) – 16 2 4(2) - 16 3 4(3) – 16 4 4(4) – 16 5 4(5) - 16 an -12 -8 -4 0 4 d. Which term of the sequence is 32? In the formula for the term, substitute 32 for an. an = 4n – 16 32 = 4n – 16 +16 +16 48 = 4n 12 = n (n, an) (1, -12) (2, -8) (3, -4) (4, 0) (5, 4) Try this one: Consider the arithmetic sequence: 3, -10, -23, -36…. a. b. c. d. Write an equation for the nth term in the sequence Find the 15th term in the sequence Graph the first five terms Which term in the sequence is -114? Arithmetic Sequences as Functions Example: Marisol is mailing invitations to her quinceanera. The arithmetic sequence $0.42, $0.84, $1.26, $ 1.68, … represents the cost of postage. a. Write a function to represent this sequence The first term a1 is $0.42. Find the common difference 0.42 0.84 1.26 1.68 +0.42 +0.42 +0.42 The common difference is 0.42. an = a1 + (n-1)d = 0.42 + (n-1)(0.42) = 0.42 + 0.42n – 0.42 = 0.42n The function is f(n) = 0.42n b. Graph the function and determine the domain The rate of change is 0.42. Make a table and plot some points. n f(n) 1 0.42 2 0.84 3 1.26 4 1.68 5 2.10 The domain of the function is the number of invitations Marisol mails. So the domain is {1, 2, 3, 4, …} the set of natural numbers, N