1. Geometric Sequences

2. What is a Geometric Sequence?
In a geometric sequence, the ratio between consecutive terms is constant. This ratio is called the common ratio.
Unlike in an arithmetic sequence, the difference between consecutive terms varies.
We look for multiplication to identify geometric sequences.

3. Ex: Determine if the sequence is geometric
Ex: Determine if the sequence is geometric. If so, identify the common ratio
1, -6, 36, -216
yes. Common ratio=-6
2, 4, 6, 8
no. No common ratio

4. Important Formulas for Geometric Sequence:
Recursive Formula
Explicit Formula
an = (an – 1 ) r
an = a1 * r n-1
Where:
an is the nth term in the sequence
a1 is the first term
n is the number of the term
r is the common ratio

5. Ex: Write the recursive formula for each sequence
First term: a1 = 7
Common ratio = 1/3
Recursive:
an = an-1 * r
an = an-1 * (1/3)
a1 = 7
a2 = 7(1/3) = 7/3
a3 = 7/3(1/3) = 7/9
a4 = 7/9(1/3) = 7/27
a5 = 7/27(1/3) = 7/81
Now find the first five terms:

6. Ex: Write the explicit formula for each sequence
First term: a1 = 7
Common ratio = 1/3
Explicit:
an = a1 * r n-1
a1 = 7(1/3) (1-1) = 7
a2 = 7(1/3) (2-1) = 7/3
a3 = 7(1/3) (3-1) = 7/9
a4 = 7(1/3) (4-1) = 7/27
a5 = 7(1/3) (5-1) = 7/81
Now find the first five terms:

7. Recursive Geometic Sequence Problem
Find the 5th and 6th term in the sequence of 11,33,99,
an = an-1 * r
Start with the recursive sequence formula
Find the common ratio between the values.
Common ratio = 3
a5 = 297 (3) = 891
Plug in known values
a6 = 891(3) = 2673
Simplify

8. Explicit Geometic Sequence Problem
Find the 19th term in the sequence of ,33,99,
an = a1 * r n-1
Start with the explicit sequence formula
Find the common ratio between the values.
Common ratio = 3
a19 = 11 (3) (19-1)
Plug in known values
a19 = 11(3)18 =4,261,626,379
Simplify

9. Find the 10th term in the sequence of 1, -6, 36, -216 . . .
Let’s try one
Find the 10th term in the sequence of , -6, 36,
an = a1 * r n-1
Start with the explicit sequence formula
Find the common ratio between the values.
Common ratio = -6
a10 = 1 (-6) (10-1)
Plug in known values
a10 = 1(-6)9 = -10,077,696
Simplify