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Geometric Sequences

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Definition of a geometric sequence. An geometric sequence is a sequence in which each term after the first is found by multiplying the previous term by a constant called the common ratio, r.

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You can name the terms of a geometric sequence using a 1, a 2, a 3, and so on If we define the nth term as a n then then previous term is a n-1. And by the definition of a geometric sequence a n = r(a n-1 ) now solve for r r = a n a n-1

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Example 1. Find the next two terms in the geometric sequence 3, 12, 48,... First find the common ratio. Let 3 = a n-1 and let 12 = a n. r = a n a n-1 = 12 3 = 4 The common ratio is 4.

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Example 1. Geometric sequence of 3, 12, 48, … find a 4 and a 5. The common ratio is 4. a 4 = r(a 3 ) = 4(48) = 192 a 5 = r(a 4 ) = 4(192) = 768 The next two terms are 192 and 768

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There is a pattern in the way the terms of a geometric sequence are formed.

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Let’s look at example 1. numerical symbols a1a1 a2a2 a3a3 a4a4 anan In terms of r and the previous term a 1 = a 1 a 2 = r a 1 a 3 = r a 2 a 4 = r a 3

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In terms of r and the first term numerical a 1 = 3(4 0 ) a 2 = 3(4 1 ) a 3 = 3(4 2 ) a 4 = 3(4 3 ) symbols a 1 = a 1 (r 0 ) a 2 = a 1 (r 1 ) a 3 = a 1 (r 2 ) a 4 = a 1 (r 3 )

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Formula for the nth term of a geometric sequence. The nth term of a geometric sequence with first term a 1 and common ratio r is given by or a n = a n-1 r a n = a 1 r n-1

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Example 2. Write the first six terms of a geometric sequence in which a 1 = 3 and r = 2 Method 1 use a n = a n-1 (r) a 1 = 3 a 2 = 32 = 6 a 3 = 62 = 12 a 4 = 122 = 24 a 5 = 242 = 48 a 6 = 482 = 96

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Example 2. Write the first six terms of a geometric sequence in which a 1 = 3 and r = 2 Method 2 use a n = a 1 (r n-1 ) a 1 = = 3 a 2 = = 6 a 3 = = 12 a 4 = = 24 a 5 = = 48 a 6 = = 96

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Example 3. Find the ninth term of a geometric sequence in which a 3 = 63 and r = -3. Method 1 use the common ratio and the given term. a 4 = a 3 (-3) = 63(-3) = -189 a 5 = a 4 (-3) = (-189)(-3) = 567

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Example 3. Ninth term with a 3 = 63 and r = -3 Method 1 use the common ratio and the given term. a 4 = a 3 (-3) = 63(-3) = -189 a 5 = a 4 (-3) = (-189)(-3) = 567 a 6 = a 5 (-3) = (567)(-3) = a 7 = a 6 (-3) = (-1701)(-3) = 5103

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Example 3. Ninth term with a 3 = 63 and r = -3 Method 1 use the common ratio and the given term. a 6 = a 5 (-3) = (567)(-3) = a 7 = a 6 (-3) = (-1701)(-3) = 5103 a 8 = a 7 (-3) = (5103)(-3) = a 9 = a 8 (-3) = (-15309)(-3) = 45927

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Example 3. Ninth term with a 3 = 63 and r = -3 Method 2 find a 1 a n = a 1 (r n-1 ) a 3 = a 1 (r 3-1 ) 63 = a 1 (-3) (2) 63 = a 1 (9) a 1 = 93/9 = 7 a 9 = a 1 (r (9-1) ) = 7(-3) 8 = 45927

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The terms between any two nonconsecutive terms of a geometric sequence are called the geometric means. In the sequence 3, 12, 48, 192, 769,... 12, 48, and 192 are the three geometric means between 3 and 769

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Example 4. Find the three geometric means between 3.4 and Use the nth term formula to find r. 3.4, ____, ____, ____, is a is a 5

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Example 4. Find the three geometric means between 3.4 and Use the nth term formula to find r. 3.4, ____, ____, ____, is a is a 5 a n = a 1 (r n-1 ) a 5 = 3.4(r 4 ) 2125 = 3.4(r 4 ) 625 = (r 4 ) r = 5

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Example 4. Three geometric means between 3.4 and is a is a 5 r = 5 Check both solutions If r = 5 a 2 = 3.4(5) = 17 a 3 = 17(5) = 85 a 4 = 85(5) = 425 a 5 = 425(5) = 2125

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Example 4. Three geometric means between 3.4 and is a is a 5 r = 5 Check both solutions If r = -5 a 2 = 3.4(-5) = -17 a 3 = -17(-5) = 85 a 4 = 85(-5) = -425 a 5 = -425(-5) = 2125 Both solutions check

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Example 4. Three geometric means between 3.4 and is a is a 5 r = 5 Both solutions check There are two sets of geometric means between 3.4 and , 85, and 425and -17, 85, and -425

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