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12.2 Geometric Sequences and Series. 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.
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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, a1, is the product of the preceding term and the common ratio, r. • a1, a1r, a1r2, …
Ex 1Find r and the next three terms. • A. 21, 4.2, 0.84, … • B. 2t – 10, -4t + 20, 8t – 40, …
Recursive formula for geometric sequences: an = an-1 x rn-1 • The nth term of a geometric sequence is given by: an = a1rn-1
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)
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.
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.
A geometric series is the sum of the terms of a geometric sequence.
Ex 4find the sum of the first 8 terms of 14 – 70 + 350 – 1750 + …
Banks use compound interest to determine earnings in accounts or how much to charge for loans.
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?
