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12.2 Geometric Sequences and Series

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The ration of successive terms in a geometric sequence is a constant called the common ratio, denoted r. Geometric Sequence – a sequence in which each term after the first, a 1, is the product of the preceding term and the common ratio, r. a 1, a 1 r, a 1 r 2, …

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Ex 1 Find r and the next three terms. A. 21, 4.2, 0.84, … B. 2t – 10, -4t + 20, 8t – 40, …

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Recursive formula for geometric sequences: a n = a n-1 x r n-1 The nth term of a geometric sequence is given by: a n = a 1 r n-1

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Ex 2 Find the 12 th term of -24, 26.4, -29.04, …

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Geometric sequences can model growth or decay. For a common ratio greater than 1, a sequence may model growth. (compound interest, population growth, etc.) For a positive common ratio less than 1, a sequence may model decay. (radioactive behavior and depreciation)

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Ex 3 A new car costing $23,000 depreciates at the rate of 40% per year for four years. Find the value of the car at the end of four years.

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The terms between any two nonconsecutive terms of a geometric sequence are called geometric means. Find a sequence that has two geometric means between 128 and 54.

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A geometric series is the sum of the terms of a geometric sequence.

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Ex 4 find the sum of the first 8 terms of 14 – 70 + 350 – 1750 + …

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Banks use compound interest to determine earnings in accounts or how much to charge for loans.

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Ex 5 On April 1 of every year for 25 years, Jim deposits $2000 in an IRA which pays an APR of 10% compounded annually. If he makes no withdrawals, how much will he have in the account at the end of 25 years?

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