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Arithmetic Sequences

Geometric Sequences

Arithmetic Series

Geometric Series

Sigma Notation

Mathematical Induction

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 UNIT 10 : 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 t1 , t2 , t3 , 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. t6 = 32 x 2 = 64 etc.

        The formula for the general term or nth term is tn = arn-1.

        Geometric sequences are exponential functions with domain the natural numbers N = {1, 2, 3, 4, }

Text Box: a = t1 = first term
r = common ratio
n = term number
 

 

 

 

 

 

 

 


Definition: The sum of the terms of a Geometric sequence is a Geometric Series.

Text Box: n = the number of terms
t1 = a = the first term
r = the common ratio
Sn = the sum of the series

Text Box:
 

 

 

 

 

 

 


Example 1:

Form the geometric sequence determined by the exponential function f(n) = 3(2)n-1. Find a, r and S10.

Solution:

 

a = 3

r = 2

n = 10

S10 = ?

 

 

 

a = - 2

r = 3

n = 9

S9 = ?

 
Example 2: Given the first few terms.

Given the series 2 6 18 54

a) Show that the series is geometric.

b) Find S9

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 tn.

a = 2

r = 4

n = ?

tn = 32768

Sn = ?

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Example 4: When r is negative or fractional.

a) Given the geometric series 3 + 6 12 + 24 Find a, r and S12.

Solution:

 

 

Infinite Geometric Series:

 

 

 

 

 

 

 

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