Download Geometric Sequences and Series

Sections 11.3 and 11.5
Review Terms
 Sequence
 An ordered list of numbers
 Series
 The sum of the terms of a sequence
 Term
 A specific number in a sequence
 Arithmetic Sequence
 A sequence of numbers where the difference between
consecutive terms is constant
 Geometric Sequence
 A sequence of numbers where the ratio between consecutive
terms is constant
Geometric Equations
Recursive
Closed (or explicit)
This equation refers to other
terms in the sequence.
This equation allows you to find
any term in the sequence
directly.
an = an–1 ∙ r
an = a1 ∙r n–1
Geometric sequence
Determine if the following sequence is geometic. If it is,
write both types of formulas.
1. 1, –2, 4, –8, …
Yes this is. an = an–1∙ (–2) and an = (–2)n–1
2. 1, 2, 3, 4, …
No this is not geometric. The ratios keep changing.
Practice
Determine if each of the following sequences is
geometric. If it is write both types of formulas.
3. 7, 0.7, 0.07, 0.007, …
Yes. an = an–1•(0.1) or an = 7 (0.1)n–1
4. 10, 15, 22.5, 33.75, …
Yes. an = an–1 •(1.5) or an = 10 (1.5)n–1
5. 1/2, 1/4, 1/6, 1/8, …
No, there is not a common ratio.
Finite Geometric Series
This is used for finite geometric series.
n is the number of terms
a1 is the first term in the sequence
r is the common ratio between consecutive terms
Finite Series Practice
Evaluate the following series for the given number of terms:
6. 1 + 2 + 4 + …; S8
S8 = (1 (1 – 28))/(1 – 2) = 255
Finite Series Practice
Evaluate the following series for the given number of terms:
6. 1 + 2 + 4 + …; S8
S8 = (1 (1 – 28))/(1 – 2) = 255
7.
S5 = (7 (1 – (– 5)5))/(1 – (–5)) = 3647
Infinite Geometric Series
 This is used for infinite geometric series
 The variables are the same as for the finite series
 This can be used to convert repeating decimals to fractions
Infinite Series Practice
Evaluate the following geometric series, or find the fraction
equivalent for the given infinite repeating decimal.
8.
1.22222…
9. 0.222222…
a1 = 0.2, r = 0.1
S = 2/9
Practice
Convert the following infinite repeating decimals to fractions.
10. 0.42857142857142…
3/
7
11. 0.066666666…
1/
15
12. 0.2727272727…
3/
11