1. Geometric Sequences & Series
Pre-Calculus Section

2. The number that is multiplying is called the common ratio (r).
It is a “Geometric Sequence” when each term is multiplied by the same number to get the next term.
The number that is multiplying is called the common ratio (r).
To find the common ratio, divide any two consecutive terms.
(a2 ÷ a1) or (a3 ÷ a2)
1) 16, 8, 4, ...
2) 1, - 3, 9, - 27,...
r = ___
1/2
(8 ÷ 16)
r = ___
- 3
(- 3 ÷ 1)
next term = __ 2
(4 • 1/2)
next term = __ 81

3. Formula to find Any Term

4. Decide if each series is a geometric series.
If it is, state the common ratio. 1)
Yes, ratio = 2. 2)
Yes, ratio = 2/7. 3)
No common ratio. 4)
Yes, ratio = 1/2.

5. Find the next 4 terms in each sequence.
1. a1 = 6, r = 1/3
a2 = ___
a3 = ___
a4 = ___
a5 = ___ 2
2/3
2. a1 = 4, r = 5/2
2/9
a2 = ___
a3 = ___
a4 = _____
a5 = ______ 10
2/27 25
125/2
625/4

6. an = a1rn-1 In the geometric sequence where a1 = 3, r = - 2, find a10
Formula to find a specific term:
an = a1rn-1
an: any term
a1: 1st term.
r : common ratio
n: number of terms.
In the geometric sequence where a1 = 3, r = - 2, find a10
a10 =
3 •
(- 2)9 =
- 1536

7. For each geometric series, find the specific term:
a1 = 8, r = 1/2, find a11.
a11 = 1
128 8
128
a1 = - 12, r = 1/4, find a8 . 3 - 3
4096
4096

8. Find the first term. 1)
a1 = -1/2

9. Rule:
an = (2)8n-1

10. Wow…ugly number!

11. 1)

12. Geometric Series

14. an = a1rn-1 Find the sum of the following series:
-12 + (-18) + (-27) + …+ ( )
Use decimals because the last term is given as a decimal.
an = a1rn-1
We need to know how many terms are being added, so we must find the number of the last term.
n = 6
It’s the 6th term.

15. Find the sum of the following series:
-12 + (-18) + (-27) + …+ ( )
a1 = -12
n = 6
r = 1.5

16. Find the sum: a1 = 0.01(3)0 = 0.01 a2 = 0.01(3)1 = 0.03 a3 = 0.01(3)2
Find the first three terms to determine if it is arithmetic or geometric.
You will need to know the first term.
You will need to know the common ratio or common difference.
a1 = 0.01(3)0
= 0.01
The series is geometric.
a2 = 0.01(3)1
= 0.03
a1 = 0.01
r = 3
a3 = 0.01(3)2
= 0.09

17. Find the sum:
a1 = 0.01
r = 3
n = 13

18. Formulas to Memorize
Write the formulas at the beginning of each problem. You will have them memorized by the time you finish the homework.

19. Homework
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