1. Geometric Sequences

2. Definition of a geometric
sequence.
An geometric sequence is a
sequence in which each term after
the first is found by multiplying
the previous term by a constant
called the common ratio, r.

3. You can name the terms of a
geometric sequence using a1, a2, a3,
and so on
If we define the nth term as an then
then previous term is an-1.
And by the definition of a geometric
sequence an = r(an-1)
now solve for r
r = an
an-1

4. Example 1. Find the next two terms
in the geometric sequence
3, 12, 48, ...
First find the common ratio.
Let 3 = an-1 and let 12 = an.
r = an
an-1 = 12 3
= 4
The common ratio is 4.

5. Example 1. Geometric sequence of
3, 12, 48, … find a4 and a5.
The common ratio is 4.
a4 = r(a3) = 4(48) = 192
a5 = r(a4) = 4(192) = 768
The next two terms are 192 and 768

6. There is a pattern in the way the
terms of a geometric sequence
are formed.

7. Let’s look at example 1. 3 12 48
192
numerical
symbols a1 a2 a3 a4 an
In terms of r and the previous term
a2 = r • a1
a1 = a1
a3 = r • a2
a4 = r • a3

8. In terms of r and the first term
numerical
a2 = 3(41)
a1 = 3(40)
a3 = 3(42)
a4 = 3(43)
symbols
a2 = a1(r1)
a1 = a1(r0)
a3 = a1(r2)
a4 = a1(r3)

9. Formula for the nth term of a
geometric sequence.
The nth term of a geometric
sequence with first term a1 and
common ratio r is given by
an = an-1 • r or
an = a1 • rn-1

10. Example 2.
Write the first six terms of a
geometric sequence in which
a1 = 3 and r = 2
Method 1 use an = an-1(r)
a3 = 6•2 = 12
a1 = 3
a2 = 3•2 = 6
a5 = 24•2 = 48
a4 = 12•2 = 24
a6 = 48•2 = 96

11. Example 2.
Write the first six terms of a
geometric sequence in which
a1 = 3 and r = 2
Method 2 use an = a1(rn-1)
a1 = 3•21-1 = 3
a2 = 3•22-1 = 6
a4 = 3•24-1 = 24
a3 = 3•23-1 = 12
a5 = 3•25-1 = 48
a6 = 3•26-1 = 96

12. Example 3.
Find the ninth term of a geometric
sequence in which a3 = 63 and
r = -3.
Method 1 use the common ratio
and the given term.
a4 = a3(-3) = 63(-3) = -189
a5 = a4(-3) = (-189)(-3) = 567

13. Example 3. Ninth term with a3 = 63
and r = -3
Method 1 use the common ratio
and the given term.
a4 = a3(-3) = 63(-3) = -189
a5 = a4(-3) = (-189)(-3) = 567
a6 = a5(-3) = (567)(-3) = -1701
a7 = a6(-3) = (-1701)(-3) = 5103

14. Example 3. Ninth term with a3 = 63
and r = -3
Method 1 use the common ratio
and the given term.
a6 = a5(-3) = (567)(-3) = -1701
a7 = a6(-3) = (-1701)(-3) = 5103
a8 = a7(-3) = (5103)(-3) =
a9 = a8(-3) = (-15309)(-3) = 45927

15. Example 3. Ninth term with a3 = 63
and r = -3
Method 2 find a1
a3 = a1(r3-1)
an = a1(rn-1)
63 = a1(-3)(2)
63 = a1(9)
a1 = 93/9 = 7
a9 = a1(r(9-1))
= 7(-3)8 = 45927

16. The terms between any two
nonconsecutive terms of a
geometric sequence are called the
geometric means.
In the sequence
3, 12, 48, 192, 769, ...
12, 48, and 192 are the three
geometric means between 3 and 769

17. Example 4.
Find the three geometric means
between 3.4 and 2125.
Use the nth term formula to find r.
3.4, ____, ____, ____, 2125
3.4 is a1
2125 is a5

18. Example 4.
Find the three geometric means
between 3.4 and 2125.
Use the nth term formula to find r.
3.4, ____, ____, ____, 2125
3.4 is a1
2125 is a5
a5 = 3.4(r4)
an = a1(rn-1)
2125 = 3.4(r4)
625 = (r4)
r = 5

19. Example 4. Three geometric means
between 3.4 and 2125
r = 5
3.4 is a1
2125 is a5
Check both solutions
If r = 5
a2 = 3.4(5) = 17
a3 = 17(5) = 85
a4 = 85(5) = 425
a5 = 425(5) = 2125

20. Example 4. Three geometric means
between 3.4 and 2125
r = 5
3.4 is a1
2125 is a5
Check both solutions
If r = -5
a2 = 3.4(-5) = -17
a4 = 85(-5) = -425
a3 = -17(-5) = 85
a5 = -425(-5) = 2125
Both solutions check

21. Example 4. Three geometric means
between 3.4 and 2125
r = 5
3.4 is a1
2125 is a5
Both solutions check
There are two sets of geometric
means between 3.4 and 2125.
17, 85, and 425
and
-17, 85, and -425