Post on 20-Jan-2016
Geometric Sequences & Series
This chapter focuses on how to use
find terms of a geometric sequence
or series, find the sum of finite and
infinite geometric series and apply
the rules for geometric sequences
and series to problems of growth
and decay.
Geometric Sequences & Series
CONTENTS:
What is a geometric sequence?
First term and common ratio
Example 1
Nth term of a sequence or series
Example 2
Example 3
Assignment
What is a Geometric Sequence or Series?
A sequence or series where we get from one
term to the next by multiplying the
previous term by a constant number is
called a geometric sequence or series.
These are examples of geometric sequences:
(i)3, 9, 27, 81, ... Multiply by 3
(ii)16, 8, 4, 2, 1, 0.5, ...Multiply by ½
(iii)1, -2, 4, -8, 16, ... Multiply by -2
Geometric Sequences & Series
First Term
The first of a geometric sequence or series is
usually called “a”.
Common Ratio
The constant number that each term is
multiplied by is called the common ratio.
To find the common ratio we take two terms
and divide one by the previous one.
Geometric Sequences & Series
Example 1:Find the common ratio of this sequence:100, 25, 6.25, 1.5625, ...
Solution:To get the common ratio we divide a term of the sequences by the previous term. If we take the first two terms of the above sequence we get:
25/100 = ¼
Therefore r, the common ratio is ¼
Geometric Sequences & Series
nth term of a Geometric Sequence or Series
If given the first term and the common ratio
we can find any term of a geometric
sequence or series by using the formula:
nth term = arn-1
Therefore,
First term = a Second term = ar
Third term = ar2 Fourth term = ar3 ...
Geometric Sequences & Series
Example 2:Find the 10th term of the following geometric sequence:2, 6, 18, 54, ...
Solution:Begin by writing down the values of a and r.a = 2 r = 6/2 = 3
To get any term we substitute into the general formula: arn-1 where n is the number of the term we want.
10th term: ar10-1 = ar9 = 2(3)9 = 39366
Geometric Sequences & Series
Example 3: The second term of a geometric sequence is 4 and the 4th term is 16. Find:(a) common ratio (b) first term
Solution:In this question we are trying to find two unknown variables, r and a. If we have two unknown variables we need to have two equations to solve.
Equation 1: 2nd term = 4This gives: ar = 4
continued on next slide
Geometric Sequences & Series
Solution continued:Equation 2:4th term = 16This gives: ar3 = 16
Because our equations involve terms multiplied together we need to divide the equations to eliminate one of the variables.
Divide (2) by (1) to get:ar3/ar = 16/4r2 = 4r = 2
continued on next slide
Geometric Sequences & Series
Solution continued:We can find the value of a by substituting the value of r into either equation. Substituting into Equation 1 gives:
a(2) = 4a = 2
Geometric Sequences & Series
AssignmentThis weeks assignment is a Moodle Activity. There are 5 questions to answer. Please be sure to include your working out for all questions.
Deadline is 5:00pm on Monday 15th March.
Geometric Sequences & Series