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Arithmetic and Geometric Sequences (11.2) Common difference Common ratio Slide 2 A sequence Give the next five terms of the sequence for 2, 7, 12, 17, What is the pattern for the terms? Slide 3 A sequence Give the next five terms of the sequence for 2, 7, 12, 17, 22, 27, 32, 37, 42 This is an example of a sequence a string of numbers that follow some pattern. Whats our pattern here? Slide 4 A sequence Give the next five terms of the sequence for 2, 7, 12, 17, 22, 27, 32, 37, 42 Whats our pattern here? We add five to a term to get the next term. When we add or subtract to get from one term to the next, thats an arithmetic sequence. Slide 5 Another sequence Find the next five terms in this sequence? 8, 4, 2, Whats our pattern this time? Slide 6 Another sequence Find the next five terms in this sequence? 8, 4, 2, 1,.5,.25,.125,.0625 Whats our pattern this time? We divide each term by 2 to get the next term. (This is also multiplying by .) When we multiply or divide to get the next term, we have a geometric sequence. Slide 7 Terminology We label terms as t n, where n is the place the term has in the sequence. The first term of a sequence is t 1. So in the arithmetic sequence, t 1 = 2. In the geometric sequence, t 1 = 8. Slide 8 Terminology We label terms as t n, where n is the place the term has in the sequence. The second term of a sequence is t 2. The third is t 3. Get it? If the current term is t n, then the next term is t n+1. The previous term is t n-1. Slide 9 Terminology We list sequences in the abstract as t 1, t 2, t 3, t n. This is true whether the sequence is arithmetic, geometric, or neither. Slide 10 Arithmetic sequence formula If the pattern between terms in a sequence is a common difference, the sequence is arithmetic, and we call that difference d. t n = t 1 + (n-1) d (In other words, find the n th term by adding (n-1) ds to the first term.) Test it with our first sequence. Slide 11 Arithmetic sequence formula If the pattern between terms in a sequence is a common difference, the sequence is arithmetic, and we call that difference d. t n = t 1 + (n-1) d We can use this to find the first term, n th term, the number of terms, and the difference. Slide 12 Geometric sequence formula If the pattern between terms in a sequence is a common ratio, then it is a geometric sequence and we call that ratio r. t n = t 1 r n-1 (In other words, find the n th term by multiplying t 1 by r and do that (n-1) times.) Test it with our second sequence. Slide 13 Geometric sequence formula If the pattern between terms in a sequence is a common ratio, then it is a geometric sequence and we call that ratio r. t n = t 1 r n-1 We can use this to find the first term, the n th term, the number of terms, and the common ratio. Slide 14 Sequence #3 Give the first five terms of the sequence for t 1 = 7 t n+1 = t n 3 What is the pattern for the terms? Is this arithmetic or geometric? What is the tenth term? Slide 15 Sequence #3 Give the first five terms of the sequence for 7, 4, 1, -2, -5 What is the pattern for the terms? We subtract 3 from a term to get the next one. It is an arithmetic sequence. The tenth term is t 10 = 7 + (10-1) (-3) = -20. Slide 16 Sequence #4 Find which term 101 is in the arithmetic sequence with t 1 = 5, and d = 3. Slide 17 Sequence #4 Find which term 101 is in the arithmetic sequence with t 1 = 5, and d = 3. 101 = 5 + (n 1)3 101 = 5 + 3n 3 101 = 2 + 3n 99 = 3n n = 33 So, the 33 rd term. Slide 18 Sequence #5 Find the 9 th term of the sequence 1, -2, 4, What type of sequence is this? What formula do we use? Slide 19 Sequence #5 Find the 9 th term of the sequence 1, -2, 4, What type of sequence is this? Geometric, with a common ratio of -2. What formula do we use? t n = t 1 r n-1 So, t 9 = 1(-2) 9-1 = (-2) 8 = 256. Slide 20 Sequence #6 Find which term 1536 is in the geometric sequence with t 1 = 3, and a common ratio of 2. Slide 21 Sequence #6 Find which term 1536 is in the geometric sequence with t 1 = 3, and a common ratio of 2. 1536 = 3(2) n-1 512 = (2) n-1 (Ooh, want an exponent, need to use logs.) n -1 = log 2 512 = log 512/ log2 = 9 n = 10 Slide 22 Sequence #Last Find the 9 th term of the sequence 1, 1, 2, 3, 5, 8, What type of sequence is this? What formula do we use? How do we graph it?