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**Arithmetic and Geometric Sequences (11.2)**

Common difference Common ratio

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**A sequence Give the next five terms of the sequence for**

2, 7, 12, 17, … What is the pattern for the terms?

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**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. What’s our pattern here?

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**A sequence Give the next five terms of the sequence for**

2, 7, 12, 17, 22, 27, 32, 37, 42 What’s 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, that’s an arithmetic sequence.

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**Another sequence Find the next five terms in this sequence? 8, 4, 2, …**

What’s our pattern this time?

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**Another sequence Find the next five terms in this sequence?**

8, 4, 2, 1, .5, .25, .125, .0625 What’s our pattern this time? We divide each term by 2 to get the next term. (This is also multiplying by ½.) When we multiply or divide to get the next term, we have a geometric sequence.

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Terminology We label terms as tn, where n is the place the term has in the sequence. The first term of a sequence is t1. So in the arithmetic sequence, t1 = 2. In the geometric sequence, t1 = 8.

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Terminology We label terms as tn, where n is the place the term has in the sequence. The second term of a sequence is t2. The third is t3. Get it? If the current term is tn, then the next term is tn+1. The previous term is tn-1.

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**Terminology We list sequences in the abstract as t1, t2, t3, … tn.**

This is true whether the sequence is arithmetic, geometric, or neither.

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**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. tn = t1 + (n-1) d (In other words, find the nth term by adding (n-1) d’s to the first term.) Test it with our first sequence.

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**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. tn = t1 + (n-1) d We can use this to find the first term, nth term, the number of terms, and the difference.

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**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. tn = t1rn-1 (In other words, find the nth term by multiplying t1 by r and do that (n-1) times.) Test it with our second sequence.

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**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. tn = t1rn-1 We can use this to find the first term, the nth term, the number of terms, and the common ratio.

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**Sequence #3 Give the first five terms of the sequence for t1 = 7**

tn+1 = tn – 3 What is the pattern for the terms? Is this arithmetic or geometric? What is the tenth term?

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**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 t10 = 7 + (10-1) (-3) = -20.

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Sequence #4 Find which term 101 is in the arithmetic sequence with t1 = 5, and d = 3.

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Sequence #4 Find which term 101 is in the arithmetic sequence with t1 = 5, and d = 3. 101 = 5 + (n – 1)3 101 = 5 + 3n – 3 101 = 2 + 3n 99 = 3n n = 33 So, the 33rd term.

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**Sequence #5 Find the 9th term of the sequence 1, -2, 4, …**

What type of sequence is this? What formula do we use?

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**Sequence #5 Find the 9th 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? tn = t1rn-1 So, t9 = 1(-2)9-1 = (-2)8 = 256.

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Sequence #6 Find which term 1536 is in the geometric sequence with t1 = 3, and a common ratio of 2.

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Sequence #6 Find which term 1536 is in the geometric sequence with t1 = 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 = log2512 = log 512/ log2 = 9 n = 10

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**Sequence #Last Find the 9th 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?

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