Geometric Sequences
and Series
Section 8.3 beginning on page 426
Geometric Sequences
In a geometric sequence, the ratio of any term to the previous term is constant. This
constant ratio is called the common ratio and is denoted by r.
Example 1: Tell whether each sequence is geometric.
a) 6, 12, 20, 30, 42,…
12 20
6 12
2
5
3
Since the ratios are
different, this is not a
geometric sequence.
b) 256, 64, 16, 4, 1,…
64 16 4
256 64 16
1
4
1
4
1
4
1
4
1
4
Because we have a
common ratio, this
sequence is geometric.
Rules for Geometric Sequences
The nth term of a geometric sequence with first term 𝑎1 and common ratio 𝑟 is given by
the following formula.
𝑎𝑛 = 𝑎1 𝑟 𝑛−1
For example, a geometric sequence with a first term of 2 and a common ratio of 3 would
have the following formula.
𝑎𝑛 = 2(3)𝑛−1
Writing Rules
Example 2: Write a rule for the nth term of each sequence. Then find 𝑎8 .
a) 5, 15, 45, 135,…
15 45 135
5 15 45
3
3
3
𝑎𝑛 = 𝑎1 𝑟 𝑛−1
𝑎𝑛 = 5(3)𝑛−1
𝑎8 = 5(3)8−1
𝑎8 = 5(3)7
𝑎8 = 10,935
b) 88, -44, 22, -11,…
−44 22 −11
88 −44 22
1
1
1
−
−
−
2
2
2
𝑎𝑛 = 𝑎1 𝑟 𝑛−1
1
𝑎𝑛 = 88 −
2
𝑛−1
1
𝑎8 = 88 −
2
8−1
1
𝑎8 = 88 −
2
7
11
𝑎8 = −
16
Writing a Rule Given a Term and
the Common Ratio.
Example 3: One term or a geometric sequence is 𝑎4 = 12. The common ratio is 𝑟 = 2.
Write a rule for the nth term. Then graph the first 6 terms of the sequence.
𝑎𝑛 = 𝑎1 𝑟 𝑛−1
𝑎𝑛 = 𝑎1 𝑟 𝑛−1
𝑎𝑛 = 1.5(2)𝑛−1
𝑎4 = 𝑎1 (2)4−1
𝑎𝑛 = 1.5(2)𝑛−1
𝑎2 = 1.5(2)1
=3
12 = 𝑎1 (2)3
𝑎3 = 1.5(2)2
=6
12 = 𝑎1 ∙ 8
𝑎4 = 1.5(2)3
= 12
1.5 = 𝑎1
𝑎5 = 1.5(2)4
= 24
𝑎6 = 1.5(2)5
= 48
Writing a Rule Given Two Terms
Example 4: Two terms of geometric sequence are 𝑎2 = 12 and 𝑎5 = −768. Write a rule
for the nth term.
*Create two equations using the given info.
𝑎5 = 𝑎1 𝑟 5−1
𝑎2 = 𝑎1 𝑟 2−1
−768 = 𝑎1 𝑟 4
12 = 𝑎1 𝑟1
*Solve the system you created to find r and 𝑎1 .
12 = 𝑎1 𝑟1
−768 = 𝑎1 𝑟 4
*Write the rule using r and 𝑎1 .
𝑎𝑛 = 𝑎1
12
= 𝑎1
𝑟1
12
= 𝑎1
(−4)
12 4
𝑟
𝑟1
−768 = 12𝑟 3
−64 = 𝑟 3
𝑟 𝑛−1
𝑎𝑛 = −3(−4)𝑛−1
−768 =
−3 = 𝑎1
−4 = 𝑟
Sums of a Finite Geometric Series
𝑆𝑛 = 𝑎1
1 − 𝑟𝑛
1−𝑟
Example 5: Find the sum
10
4(3)𝑘−1
**Identify n and r and the
first term.
𝑛 = 10
𝑟=3
𝑘=1
**Plug the values in to the
formula and then use your
calculator.
𝑆10
1 − 310
=4
1−3
𝑆10 = 118,
𝑎1 = 4
Monitoring Progress
Tell whether the sequence is geometric. Explain your reasoning.
1
1) 27,9,3,1, 3
2) 2,6,24,120,720, …
1
Not geometric
Geometric, 𝑟 = 3
3) −1,2, −4,8, −16, …
Geometric, 𝑟 = −2
4) Write a rule for the nth term of the sequence 3,15,75,375, … then find 𝑎9 .
𝑎𝑛 = 3(5)𝑛−1
𝑎9 = 1,171,875
Write a rule for the nth term of the sequence and find the first three terms.
5) 𝑎6 = −96, 𝑟 = −2
𝑎𝑛 = 3(−2)𝑛−1
3, −6,12
6) 𝑎2 = 12, 𝑎4 = 3
1
𝑎𝑛 = −24 −
2
1
𝑎𝑛 = 24
2
𝑛−1
− 24,12, −6
𝑛−1
24,12,6
Monitoring Progress
Find the sum.
8
5𝑘−1
7)
7
12
𝑘=1
𝑆8 = 97,656
6(−2)𝑖−1
8)
𝑖=1
𝑆12 = −8,190
−16(0.5)𝑡−1
9)
𝑡=1
𝑆7 = −31.75