Warm up: There is an arithmetic sequence in each column and row.
Fill in all the numbers.
(We know because it's linear)
18. x = 3
19. x = 4
After looking over the following sequences, what is
your definition of a geometric sequence?
9.3 Geometric Sequences
The sequences in BLUE on the LEFT side are Geometric.
The sequences in RED on the RIGHT side are NOT Geometric.
3, 6, 12, 24, 48
1, 2, 3, 4
1, 2, 4, 8
6, 9, 12, 15, 18
18, -6, 2, -2/3
1, -2 , 6, -24
-3, 6, -12, 24
1/2, 1/4, 1/6, 1/8
Definition: A geometric sequence is a sequence in which each
term after the first term is found by multiplying the previous term by
a constant r called the common ratio.
Example 1: Find the next 2 terms of each geometric
sequence.
a. 8, 20, 50, 125, _____, _____
b. 324, 108, 36, 12, _____, _____
Example 2: Determine whether each of the following is a
geometric sequence. If it is, find the common ratio.
To derive the formula for a geometric sequence, consider the
sequence below:
2, 6, 18, 54,162, ...
a. 6, 10, 14, 18, ...
Now, express each term in terms of the common ratio,
first term, .
and the
b. 3, -6, 12, -24, 48, ...
You fill in the rest.
Do you see a pattern? Can you generalize to write an equation for
The nth term
of a geometric sequence with
first term
and common ratio is given by
where n is any positive integer.
?
Example 3: Find the 6th term of a geometric sequence for
which
and
.
Your turn: Find the 8th term of a geometric sequence for
which
and
.
Example 4: Write an equation for the n term of each
geometric sequence below.
th
a. 5, 10, 20, 40, ...
b. 3, 12, 48, 192, ...
Your turn: Find the tenth term of a geometric sequence
for which
and
.
Example 5: Find the seventh term of a geometric sequence
for which
and
.
Example 6: Given that the 5th and 8th terms of a
geometric sequence are 256 and 2048, respectively, write
a rule for the nth term of the sequence.
Finding Geometric Means
Example 7: Find the three geometric means between
3.12 and 49.92.
Here's an shortcut to find r
Your turn: Find the four geometric means between
6 and 192.
Your turn: Find the three geometric means between
2.25 and 576.
Recall
New
Explicit formula for
Arithmetic sequence
Explicit formula for
Geometric sequence
Recursive formula for
Arithmetic sequence
Recursive formula for
Geometric sequence
Example 8: Write a recursive rule for the terms in the
geometric sequences below. Then, find the next term.
a. 3, 6, 12, 24, ...
b. 10, 2, 2/5, 2/25, ...
Example 9: Using the recursive formula an = an - 1 2, find the
next three terms if a1 = 3.