Transcript Slide 1

Exercise
2, 4, 8, 16, …
32
Exercise
2, 4, 6, 8, …
10
Exercise
1, 3, 9, 27, …
81
Exercise
1 1 1
1, 2 , 4 , 8 , …
1
16
Exercise
1, –2, 4, –8, 16, …
–32
3 6 12 24 48 96
x2 x2 x2 x2 x2
Geometric Sequence
A geometric sequence is a
sequence of numbers
whose successive terms
differ by a constant
multiplier.
Common Ratio
The constant multiplier for
a geometric sequence is
called the common ratio, r.
Example
State whether the sequence
8, 4, 2, 1, … is arithmetic or
geometric.
geometric
Example
State whether the sequence
–6, –18, –54, –162, … is
arithmetic or geometric.
geometric
Example
State whether the sequence
5, 7, 9, 11, … is arithmetic or
geometric.
arithmetic
Example
State whether the sequence
5, 10, 20, 40, … is arithmetic
or geometric.
geometric
Geometric Sequence
Terms differ by a
constant factor r.
an = an – 1r
Example 1
Write the first six terms of
the geometric sequence in
which a1 = 1 and r = 3.
a1 = 1
a2 = 1 • 3 = 3
a3 = 3 • 3 = 9
a4 = 9 • 3 = 27
Example 1
Write the first six terms of
the geometric sequence in
which a1 = 1 and r = 3.
a5 = 27 • 3 = 81
a6 = 81 • 3 = 243
The first six terms of
the sequence are
1, 3, 9, 27, 81, and 243.
Example 2
Find the value of a1 for the
sequence
2, 6, 18, 54, 162, 486, …
a1 = 2
Example 2
Find the value of r for the
sequence
2, 6, 18, 54, 162, 486, …
r=3
Example 2
Find the value of a3 for the
sequence
2, 6, 18, 54, 162, 486, …
a3 = 18
Example 2
Find the value of a8 for the
sequence
2, 6, 18, 54, 162, 486, …
a7 = 486 • 3
= 1,458
a8 = 1,458 • 3
= 4,374
Geometric Sequence
Terms differ by a
constant factor r.
an = an – 1r
Example 3
Write the first five terms of
the sequence defined by
a1 = –4 and an = 3an – 1.
a1 = –4
a2 = 3(–4) = –12
a3 = 3(–12) = –36
a4 = 3(–36) = –108
a5 = 3(–108) = –324
Example 3
Write the first five terms of
the sequence defined by
a1 = –4 and an = 3an – 1.
The first five terms of
the sequence are
–4, –12, –36, –108, and
–324.
Example
Write the first four terms of
the sequence defined by
a1 = 2 and an = 4an – 1.
2, 8, 32, 128
Example
Write the first four terms of
the sequence defined by
a1 = –3 and an = 2an – 1.
–3, –6, –12, –24
Example
Find the common ratio, r, of
the sequence
4, –12, 36, –108.
r = –3
Example
Find the common ratio, r, of
the sequence 24, 12, 6, 3.
1
r= 2
Example 4
Write the recursive formula
for the sequence
729, 243, 81, 27, ...
a1 = 729
1
r= 3
1
an = 3 an – 1
3, 6 ,12, 24, 48, 96
× 2 to get next term
Position
1
2
3
4
5
6
n
Term
3
3 • 21 = 6
3 • 22 = 12
3 • 23 = 24
3 • 24 = 48
3 • 25 = 96
3 • 2n – 1
Explicit Formula
The explicit formula for a
geometric sequence is
an = a1r n–1, in which a1 is
the first term and r is the
common ratio.
Example 5
Write the explicit formula for
the sequence
–5, –15, –45, –135, –405, ...
a1 = –5
r=3
an = –5(3)n – 1
Example
Write the explicit formula for
the sequence
–3, –6, –12, –24, ...
an = –3(2)n – 1
Example
Write the explicit formula for
the sequence 12, 6, 3, 1.5, ...
1 n–1
an = 12( 2 )
Exercise
A ball bounces three-fourths
the height of its fall. If the ball
falls 12 ft., how high does it
bounce on the first bounce?
on the second bounce? on
the third bounce?
9 ft.; 6.75 ft.; 5.0625 ft.
Exercise
In the last problem, the height
of the bounces forms a
geometric sequence. Find the
common ratio of this
geometric sequence.
r = 0.75
Exercise
If the ball falls 12 ft. and
begins bouncing, what is the
total distance it has traveled
when it hits the ground the
third time?
43.5 ft.
Exercise
When will the ball stop
bouncing?