SEQUENCES AND SERIES
Section 12-2 Geometric Sequences and Series
Definition of a geometric sequence
 A geometric sequence is a sequence in which each
term after the first, a1, is the product of the
preceding term and the common ratio, r. the
terms of the sequence can be represented as
follows, where a1 is nonzero and r is not equal to 1
or 0. The common ratio divide a term by its
preceding term (order is important! a2/a1)
2
 a1,a1r,a1r ,… .
Example # 1
 Determine the common ratio and find the next three terms in each sequence.
 1. 21,4.2,0.84 …
 2. 2t-10, -4t+20, 8t -40
 #1 First find the common ratio: a2/a1= 4.2/21= .2
 Now multiply .84 (.2)=.168 and so on. Next three terms: .168, .0336, .00672
 #2 The common ratio = -4t+20/ 2t -10 = -2t+10/t-5 = -2
The next term is (8t-40)(-2)=-16t+80
Next three terms are -16t+80, 32t-160, -64t+320
The nth term of a geometric sequence
 The nth term of a geometric sequence with first
term a1 and common ratio r is given by
 an = a1r
n 1
Example # 2
 Find an approximation for the 12th term in the
sequence -24, 26.4, -29.04,… .
 Solution:
 The common ratio is 26.4/-24= -1.1
11
 a12= -24(-1.1) ≈ 68.5
Geometric means
 Geometric sequences can represent growth or decay.
 The terms between any two nonconsecutive terms of
a geometric sequence are called geometric means.
 A geometric series is the indicated sum of the terms
of a geometric sequence.
 Remember a geometric sequence is a list of terms,
each generated by a common ratio, where as a
geometric series is the indicated sum of those terms.
Example # 3
 Write a sequence that has two geometric means
between 128 and 54.
 This sequence will have the form 128, ?, ?, 54
3
 The common ratio: 54= 128r
 128 (3/4)=96
96 (3/4)= 72
27/64=r 3 r= ¾
72 (3/4) = 54
 The two missing terms are 96 and 72
Sum of a finite geometric series
 The sum of the first n terms of a finite geometric
series is given by
n
 Sn= a1 – a1r
1-r
Example # 4
 Find the sum of the first eight terms of the
geometric series 14-70+350-1750+… .
 The common ratio is -70/14= -5
8
 S8=14-14(-5) = 5468764/6 = 911460

1--5
HW # 41
Section 12-2
Pp. 771-773
#17-31 odds, 41,48,50