1. Geometric Sequences

2. Geometric Sequences The ratio between consecutive terms is a constant.
This constant ratio is called the common ratio (r).

3. Geometric Sequences
Lesson 11-3
Additional Examples
Is the given sequence geometric? If so, identify the common ratio.
a. 1, –6, 36, –216, . . .
1,           –6,           36,           –216
–6 ÷ 1 = –6
 –6
36 ÷ –6 = –6
 –6
216 ÷ 36 = –6
 –6
There is a common ratio of –6. This is a geometric sequence.

4. Geometric Sequences
Lesson 11-3
Additional Examples
(continued)
b. 2, 4, 6, 8, . . .
2,            4,             6,             8
4 ÷ 2 = 2
 2  3 2
6 ÷ 4 =  4 3
8 ÷ 6 =
There is no common ratio. This is not a geometric sequence.

5. Geometric Sequence Formula

6. an = a1 • r n – 1 Use the explicit formula.
Geometric Sequences
Lesson 11-3
Additional Examples
Suppose you have equipment that can enlarge a photo to 120% of its original size. A photo has a length of 10 cm. Find the length of the photo after 5 enlargements at 120%.
You need to find the 6th term of the geometric sequence 10, 12, 14.4, . . .
an = a1 • r n – 1 Use the explicit formula.
a6 = 10 • – 1 Substitute a1 = 10, n = 6, and r = 1.20.
= 10 • Simplify the exponent.
Use a calculator.
After five enlargements of 120%, the photo has a length of about 25 cm.

7. Geometric Mean

8. geometric mean = 150,000 • 188, 160 Use the definition.
Geometric Sequences
Lesson 11-3
Additional Examples
A family purchased a home for $150,000. Two years later the home was valued at $188,160. If the value of the home is increasing geometrically, how much was the home worth after one year?
geometric mean = ,000 • 188, 160 Use the definition.
= 28,224,000,000 Multiply.
= 168,000 Take the square root.

9. 5. Find the missing term for the geometric sequence 3, , 48 . . . 12
Geometric Sequences
Lesson 11-3
Lesson Quiz
Is the given sequence geometric? If so, identify the common ratio and find the next two terms.
1. 1, 2, 6, 12, . . .
2. 2, 1, 0.5, 0.25, . . . no
yes; 0.5; 0.125,
3. –9, 81, –729, 6561, . . .
yes; –9, –59,049, 531,441
4. Write the explicit formula for the geometric sequence for which a1 = 7 and r = . Then generate the first five terms. 1 3
an = 7 • ; 7, , , , 1 3
n – 1 7 9 27 81
5. Find the missing term for the geometric sequence 3, , 12