Download Angles, Degrees, and Special Triangles

Geometric Sequences & Series
MATH 109 - Precalculus
S. Rook
Overview
• Section 9.3 in the textbook:
– Geometric sequences
– Partial sums of finite geometric sequences
– Infinite geometric series
2
Geometric Sequences
Geometric Sequences
• Geometric sequence: a sequence where the
ratio of ANY two successive terms is equal to
the same constant value
for all natural numbers i where r is known
as the common ratio
– e.g.:1,2,4,8,,2n1 a1 = 1 and r = 2
– e.g.: 4,2,1, 1 ,,4 1  a1 = 4 and r = ½
r
ai 1
ai
n 1
2
2
4
Geometric Sequences (Continued)
• The formula for the nth term of a geometric
sequence is an  a1r n1 where a1 is the first term
of the sequence and r is the common ratio
5
Geometric Sequences (Example)
Ex 1: Find the indicated term using the given
geometric sequence:
a) 80, 20, 5, … ; 6th
b) -3, 9, -27, … ; 11th
6
Partial Sums of Finite Geometric
Sequences
Partial Sums of Finite Geometric
Sequences
• The nth partial sum of a geometric sequence is
1 r n 
given by
where
a
is
the
first
1
, r  1
S n  a1 
term and r is the
 1 r 
common ratio
– Do not need to worry about deriving the formula
– Just know how to use it
• Also known as a Finite Geometric Series
8
Partial Sums of Finite Geometric
Sequences (Example)
Ex 2: Evaluate the partial sum for the given
finite geometric sequence:
a) n = 8 where the nth term is given by
an = 7(2)n-1
b) n = 12 where the sixth term is 4⁄125 and the
seventh term is 4⁄625
9
Infinite Geometric Series
Infinite Geometric Series
• Infinite Geometric Series: a summation of
ALL the terms of an infinite geometric
sequence
• If |r| < 1, you will see in Calculus that rn will
approach zero (rn → 0) as n approaches
positive infinity (n → +oo)
• Thus, the sum of an infinite geometric series is

a1
S   a1r 
1 r
i 1
i
– Again, just know how to use the formula to solve
problems
11
Infinite Geometric Series (Example)
Ex 3: Evaluate the infinite geometric series:

1
a)  4 
n 0  4 

n
b)   3 0.9
n
n 0
12
Infinite Geometric Series (Example)
Ex 4: Find the rational number representation
of 0.36
13
Summary
• After studying these slides, you should be able to:
– Identify whether a given series is geometric
– Evaluate the partial sum of a geometric sequence
– Evaluate an infinite geometric series
• Additional Practice
– See the list of suggested problems for 9.3
• Next lesson
– Study for the Final Exam!
14