Look at the ratios between successive terms. If the ratios are all the same, the sequence is geometric. If not, it isn't.
This IS a geometric sequence.
Example 2
Determine whether the sequence is geometric.
3,6,9,12,...
Answer
Look at the ratios between successive terms. If the ratios are all the same, the sequence is geometric. If not, it isn't.
We don't have to go any further. The ratios aren't all the same, so this is NOT a geometric sequence.
Example 3
Determine whether the sequence is geometric.
10, 15, 22.5, 33.75, ...
Answer
Look at the ratios between successive terms. If the ratios are all the same, the sequence is geometric. If not, it isn't.
The ratios are all the same, so this IS a geometric sequence.
Example 4
Determine whether the sequence is geometric.
3, 12, 48, 192, ...
Answer
Look at the ratios between successive terms. If the ratios are all the same, the sequence is geometric. If not, it isn't.
This IS a geometric sequence.
Example 5
Determine whether the sequence is geometric.
.9, .8, .7, .6, ...
Answer
Look at the ratios between successive terms. If the ratios are all the same, the sequence is geometric. If not, it isn't.
The ratios aren't the same, so this is NOT a geometric sequence.
Example 6
Determine if the sequence is geometric or not.
1,-1,1,-1,...
Answer
Any ratio between successive terms is either
or
Since the ratio is always -1, this IS a geometric sequence.
Example 7
Determine if the sequence is geometric or not.
-2, 14, -98, 686, ...
Answer
Since the ratio between successive terms is always -7, this IS a geometric sequence.
Example 8
Determine if the sequence is geometric or not.
3, -15, -75, 375, ...
Answer
The ratios between successive terms are not always the same. This is NOT a geometric sequence.
Example 9
Find the first four terms of the geometric sequence with
a1 = 7 and r = 3
Answer
To get from one term to the next, multiply by 3. The first four terms are
7, 21, 63, 189
Example 10
Find the first four terms of the geometric sequence with
a1 = 100 and
Answer
To get from one term to the next, multiply by (equivalently, divide by 5).
The first four terms are
100, 20, 4, 0.8
Example 11
Find the first four terms of the geometric sequence with
a1 = 2 and r = 0.5
Answer
To get from one term to the next, multiply by 0.5 (equivalently, divide by 2).
The first four terms are
2, 1, 0.5, 0.25
Example 12
Find the first four terms of the geometric sequence with
a1 = 8 and r = -4
Answer
To get from one term to the next, multiply by -4:
8, -32, 128, -512
Example 13
Find the first four terms of the geometric sequence with
a1 = 3 and r = 1
Answer
Since we're multiplying by 1, all the terms are the same:
3, 3, 3, 3
Example 14
A geometric sequence has a1 = 4 and r = 5. Find
a) a2
b) a3
c) a4
d) an
Answer
To get from one term to the next we multiply by 5. In order to see the pattern, we recommend NOT multiplying the numbers together, but keeping the expression for each term in factored form.
a) a2 = a1(5) = 4 × 5
b) a3 = a2 × 5= 4 × 5 × 5= 4 × 52
c) a4 = a3 × 5= 4 × 52 × 5= 4 × 53
d) We haveWe conclude thatan = 4 × 5n – 1.
Example 15
A geometric sequence has a1 = 3 and . Find
a) a2
b) a3
c) a4
d) an
Answer
To get from one term to the next we multiply by . In order to see the pattern, we recommend keeping terms in factored form.
a)
b)
c)
d) We have We must have
Example 16
A general geometric sequence has first term a1 and common ratio r. Find
a) a2
b) a3
c) a4
d) an, the general formula for the nth term.
Answer
We don't have numbers for a1 and r, but other than that, nothing has changed.
a) a2 = a1r
b) a3 = a2r= a1rr= a1r2
c) a4 = a3r= a1r2r= a1r3
d) Look at the terms we have so far:This means the general formula for the nth term of a geometric sequence isan = a1rn – 1.
Example 17
A geometric sequence has a = 8 and r = 0.2. Find a5.
Answer
We have a = 8, r = 0.2 and we want to find a5. We plug the numbers into the appropriate places:
an = arn – 1
a5 = (8)(0.2)4
= 0.0128.
Example 18
A geometric sequence has a = 2 and r = -5. Find a7.
Answer
We plug the numbers into the appropriate places in the formula:
an = arn – 1
a7 = (2)(-5)6
= 31,250.
Example 19
A geometric sequence has a = 0.125 and r = 2. Find a20.
Answer
Plug in the numbers:
an = arn – 1
a20 = (0.125)(2)20
= 131,072.
Example 20
Find a and r for the geometric sequence with the given terms.
a7 = 640 and a8 = -1280
Answer
The common ratio is
Plugging this and a7 = 640 into the formula, we get
an = arn – 1
a7 = a(-2)6
640 = a(64)
We conclude that
Example 21
Find a and r for the geometric sequence with the given terms.
a4 = 1.2 and a5 = 0.24
Answer
The common ratio is
Plugging things into the formula, we get
an = arn – 1
a4 = a(0.2)3
1.2 = a(.008)
We conclude that
Example 22
Find a and r for the geometric sequence with the given terms.
a4 = -3.125 and a5 = 0.78125
Answer
The common ratio is
Using the nice formula, we get
an = arn – 1
a4 = a(-0.25)3
-3.125 = a(-0.015625)
We conclude that
Example 23
Find a and r for the geometric sequence with the given terms.
a11 = 1 and a13 = 0.0625
Answer
Find the ratio between the given terms:
Since there are 2 multiplications from a11 to a13, take the square root to get r:
r = (0.0625)1/2 = 0.25
Then use the formula to find a.
an = arn – 1
a11 = ar10
1 = a(0.25)10
We conclude that
Example 24
Find a and r for the geometric sequence with the given terms.
a4 = 51.2 and a7 = 26.2144
Answer
Find the ratio between the given terms:
Since there are 3 multiplications between these terms, take the third root to get r:
(equivalently, divide by 5).
The first four terms are 
We conclude thatan = 4 × 5n – 1.
. Find
. In order to see the pattern, we recommend keeping terms in factored form.
We must have
This means the general formula for the nth term of a geometric sequence isan = a1rn – 1.
