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13.2 Arithmetic & Geometric Sequences Today’s Date: 5/1/14

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Arithmetic sequence (defined recursively) A sequence a 1, a 2, a 3, … if there is a constant d for which a n = a n–1 + d for n > 1 (defined explicitly) the general term is a n = a 1 + (n – 1)d d is the common difference d = a n – a n–1 Ex 1) Determine if the sequence is arithmetic. If yes, name the common difference. a)20, 12, 4, –4, –12, …yesd = –8 b)9.3, 9.9, 10.5, 11.1, 11.7, …yesd = 0.6 Ex 2) Create your own arithmetic sequence with common difference of –1.5 (share a few together) Ex 3) Find the 102 nd term of the sequence 5, 13, 21, 29, … a 1 = 5a 102 = 5 + (102 – 1)(8) d = 8 = 5 + 808 = 813

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The graph of a sequence is a set of points – NOT a continuous curve If we know two terms of a sequence, we can find a formula. Ex 4) In an arithmetic sequence, a 5 = 24 and a 9 = 40. Find the explicit formula. (write what we know) 24 = a 1 + (5 – 1)d 40 = a 1 + (9 – 1)d Solve the system: a 1 + 4d = 24 a 1 + 8d = 40 – 4d = –16a 1 + 16 = 24 d = 4 a 1 = 8 a n = 8 + (n – 1)(4) or a n = 4 + 4n sequence continuous function – ––

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If a 1, a 2, a 3, …, a k–1, a k is an arithmetic sequence, then a 2, a 3, …, a k–1 are arithmetic means between a 1 and a k. Ex 5) Find 3 arithmetic means between 9 and 29. 9, ___, ___, ___, 29 *this is a stream-lined way to solve* last – first # of commas 141924 Geometric Sequence (defined recursively) A sequence a 1, a 2, a 3, …if there is a constant r for which a n = a n–1 · r for n > 1 (defined explicitly) the general term is a n = a 1 r n–1 r is the common ratio

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Ex 6) Create your own geometric sequence with common ratio r = –2. (share please) Ex 7) Find an explicit formula for the geometric sequence 4, 20, 100, 500, … and use it to find the ninth term. a 1 = 4

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If a 1, a 2, a 3, …, a k–1, a k is a geometric sequence, then a 2, a 3, …, a k–1 are called geometric means between a 1 and a k. Ex 8) Locate 3 geometric means between 4 and 324. 4, ___, ___, ___, 324 or 4, ___, ___, ___, 324 *stream-lined way to solve* # of commas 1236108 –1236–108 A single geometric mean is called the geometric mean. (the signs must be the same) Ex 9) Find the mean proportional m (if it exists) between: a) –42 and –378b) 1 and –16 DNE

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Homework #1302 Pg 687 #1–3, 11, 13, 14, 15, 18, 20, 23–25, 28–30, 32, 34, 35, 38 *There are several word problems in the homework – just make the sequence and apply the rules!

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