 # Geometric Sequences & Series

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Geometric Sequences & Series
Pre-Calculus Section

The number that is multiplying is called the common ratio (r).
It is a “Geometric Sequence” when each term is multiplied by the same number to get the next term. The number that is multiplying is called the common ratio (r). To find the common ratio, divide any two consecutive terms. (a2 ÷ a1) or (a3 ÷ a2) 1) 16, 8, 4, ... 2) 1, - 3, 9, - 27,... r = ___ 1/2 (8 ÷ 16) r = ___ - 3 (- 3 ÷ 1) next term = __ 2 (4 • 1/2) next term = __ 81

Formula to find Any Term

Decide if each series is a geometric series.
If it is, state the common ratio. 1) Yes, ratio = 2. 2) Yes, ratio = 2/7. 3) No common ratio. 4) Yes, ratio = 1/2.

Find the next 4 terms in each sequence.
1. a1 = 6, r = 1/3 a2 = ___ a3 = ___ a4 = ___ a5 = ___ 2 2/3 2. a1 = 4, r = 5/2 2/9 a2 = ___ a3 = ___ a4 = _____ a5 = ______ 10 2/27 25 125/2 625/4

an = a1rn-1 In the geometric sequence where a1 = 3, r = - 2, find a10
Formula to find a specific term: an = a1rn-1 an: any term a1: 1st term. r : common ratio n: number of terms. In the geometric sequence where a1 = 3, r = - 2, find a10 a10 = 3 • (- 2)9 = - 1536

For each geometric series, find the specific term:
a1 = 8, r = 1/2, find a11. a11 = 1 128 8 128 a1 = - 12, r = 1/4, find a8 . 3 - 3 4096 4096

Find the first term. 1) a1 = -1/2

Rule: an = (2)8n-1

Wow…ugly number!

1)

Geometric Series

an = a1rn-1 Find the sum of the following series:
-12 + (-18) + (-27) + …+ ( ) Use decimals because the last term is given as a decimal. an = a1rn-1 We need to know how many terms are being added, so we must find the number of the last term. n = 6 It’s the 6th term.

Find the sum of the following series:
-12 + (-18) + (-27) + …+ ( ) a1 = -12 n = 6 r = 1.5

Find the sum: a1 = 0.01(3)0 = 0.01 a2 = 0.01(3)1 = 0.03 a3 = 0.01(3)2
Find the first three terms to determine if it is arithmetic or geometric. You will need to know the first term. You will need to know the common ratio or common difference. a1 = 0.01(3)0 = 0.01 The series is geometric. a2 = 0.01(3)1 = 0.03 a1 = 0.01 r = 3 a3 = 0.01(3)2 = 0.09

Find the sum: a1 = 0.01 r = 3 n = 13

Formulas to Memorize Write the formulas at the beginning of each problem. You will have them memorized by the time you finish the homework.

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