
UNIT
8 :
SEQUENCES AND SERIES
LESSON 4:
GEOMETRIC SERIES
Geometric
Series:
Recall a sequence such as 2, 4, 8, 16, 32, … is called a Geometric
Sequence. These sequences have the following properties.
·
Terms
are denoted as t_{1} , t_{2} , t_{3 , }referring
_{ }to term1, term 2,
term 3 …
·
·
This
ratio is called the common ratio and denoted using the letter r. Here r = 2.
·
The
first term is denoted using the letter a. Here a = 2.
·
Successive
terms are found by multiplying a given term by the common ratio. Eg.
t_{6} = 32 x 2 = 64 etc.
·
The
formula for the general term or nth term is t_{n }= ar^{n1}.
·
Geometric
sequences are exponential functions with domain the natural numbers N
= {1, 2, 3, 4, …}
Definition: The sum of the terms of a
Geometric sequence is a Geometric Series.
Example 1:
Form the
geometric sequence determined by the exponential function f(n) = 3(2)^{n1}. Find a, r and S_{10}.
Solution:
a = 3 r = 2 n = 10 S_{10} = ?
a =  2 r = 3 n = 9 S_{9} = ?
Example 2: Given the first few
terms.
Given
the series –2 – 6 – 18 – 54 – …
a) Show
that the series is geometric.
b) Find
S_{9}
Solution:
Example 3: Given the first
and last terms.
Given
the series 2 + 8 + 32 + … + 32768.
Find the sum of the series.
Solution: First find the number of terms n.
Let the
last term be t_{n}.
a = 2 r = 4 n = ? t_{n} = 32768 S_{n} = ?
Example 4: When r is
negative or fractional.
a) Given
the geometric series –3 + 6 – 12 + 24 – …
Find a, r and S_{12}.
Solution:
Infinite
Geometric Series: