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Geometric Sequences & Series 8.3 JMerrill, 2007 Revised 2008

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Sequences A Sequence: Usually defined to be a function Domain is the set of positive integers Arithmetic sequence graphs are linear (usually) Geometric sequence graphs are exponential

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Geometric Sequences GEOMETRIC - the ratio of any two consecutive terms in constant. Always take a number and divide by the preceding number to get the ratio 1,3,9,27,81………. ratio = 3 64,-32,16,-8,4…… ratio = -1/2 a,ar,ar2,ar3……… ratio = r

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What is the ratio of 4, 8, 16, 32… 2

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What is the ratio of 27, -18, 12,-8… -2/3

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Is the Sequence 3, 8, 13, 18… A.Arithmetic B.Geometric C.Neither

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Is the Sequence 2, 5, 10, 17… A.Arithmetic B.Geometric C.Neither

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Is the Sequence 8, 12, 18, 27… A.Arithmetic B.Geometric C.Neither

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Example Write the first six terms of the geometric sequence with first term 6 and common ratio 1/3.

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Formulas for the n th term of a Sequence Geometric:an = = = =a1 * r (n-1) To get the nth term, start with the 1st term and multiply by the ratio raised to the (n-1) power n = THE TERM NUMBER

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Example Find a formula for an and sketch the graph for the sequence 8, 4, 2, 1... Arithmetic or Geometric? r = ? an = = = = a1 (r (n-1) ) an = = = = 8 * ½ (n-1)

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Using the Formula Find the 8th term of the geometric sequence whose first term is -4 and whose common ratio is -2 an = = = =a1 * r (n-1) a8 = = = =-4 * (-2) (8-1) a8 = -4(-128) = 512

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Example Find the given term of the geometric sequence if a3 = 12, a6 =96, find a11 r = ? Since a1 is unknown. Use given info an = = = = a1 * r (n-1)an = = = = a1 * r (n-1) a3 = = = = a1 * r2a6 = = = = a1 * r5 12 = = = = a1 *r296 = a1 *r5

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Example

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Sum of a Finite Geometric Series The sum of the first n terms of a geometric series is Notice – no last term needed!!!!

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Example Find the sum of the 1st 10 terms of the geometric sequence: 2,-6, 18, -54 What is n? What is a 1 ? What is r? That’s It!

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Infinite Geometric Series Consider the infinite geometric sequence What happens to each term in the series? They get smaller and smaller, but how small does a term actually get? Each term approaches 0

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Partial Sums Look at the sequence of partial sums: What is happening to the sum? It is approaching 1 0 1

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Here’s the Rule Sum of an Infinite Geometric Series If |r| < 1, the infinite geometric series a 1 + a 1 r + a 1 r 2 + … + a 1 r n + … converges to the sum If |r| > 1, then the series diverges (does not have a sum)

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Converging – Has a Sum So, if -1 < r < 1, then the series will converge. Look at the series given by Since r =, we know that the sum is The graph confirms:

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Diverging – Has NO Sum If r > 1, the series will diverge. Look at 1 + 2 + 4 + 8 + …. Since r = 2, we know that the series grows without bound and has no sum. The graph confirms:

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Example Find the sum of the infinite geometric series 9 – 6 + 4 - … We know: a1 = 9 and r = ?

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You Try Find the sum of the infinite geometric series 24 – 12 + 6 – 3 + … Since r = -½

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Example Ex: The infinite, repeating decimal 0.454545… can be written as the infinite series 0.45 + 0.0045 + 0.000045 + … What is the sum of the series? (Express the decimal as a fraction in lowest terms)

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You Try Express the repeating decimal, 0.777…, as a rational number (hint: the sum!)

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You Try, Part Deux Find the first three terms of an infinite geometric sequence with sum 16 and common ratio

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Last Example Find the following sum: What’s the first term? What’s the second term? Arithmetic or Geometric? What’s the common ratio? Plug into the formula… 12 24 2

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Can You Do It??? Find the sum, if possible, of 8

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