Introduction to Arithmetic and
Geometric Sequences
SM1 Section 6-1
Date _____________
Objective: Students will be introduced to sequences in general, and more specifically arithmetic
and geometric sequences.
Before we can introduce arithmetic and geometric sequences, we need to establish that a
sequence is merely a pattern of numbers. For example the odd numbers develop a pattern of
numbers: 1, 3, 5, 7, 9. … the pattern is “add 2”.
Practice Problems:
Describe a pattern in the sequence of numbers. Predict the next two numbers.
1.
2, 6, 18, 54, …
2.
1, 4, 9, 16, …
3.
256, 64, 16, 4, …
4.
7, 9, 13, 19, 27, …
5.
1, 1, 2, 3, 5, 8, 13, …
6.
1, 4, 7, 10, …
The ordered list of numbers 2, 4, 8, 16, 32, … is called an infinite sequence. The order makes a
difference as to which sequence it represents. For example, 32, 16, 8, 4, 2, … used the same
numbers, but the order is different and therefore represents a different sequence.
An infinite sequence can be thought of as a pairing of each number in the sequence with a
positive integer number (i.e. 1, 2, 3, 4, 5, 6, … n …).
Position:
1
2
3
4
5 … n
Value:
2
4
8 16 32 …
It’s helpful to think of a sequence as simply
a list of values in which a position is assigned.
2n
The pairing of numbers enables us to define an infinite and finite sequence as a function with the
positive integers as the domain.
1
Sequences:
An infinite sequence is a function whose domain is the set of positive integers
1, 2, 3, 4, ... .
A finite sequence is a function whose domain is the set of positive integers when n is the
last term in the sequence. 1, 2, 3, 4, ... n
{
}
{
}
Given the sequence 2, 4, 8, 16, 32, …, we say that each number is a term of the sequence.
n
Because a sequence is a function, we could describe it by writing f (n) = 2 .
a1 = 21 = 2
First term of the sequence
a2 = 2 2 = 4
Second term
a3 = 2 = 8
Third term
3
The nth term of the sequence
a4 = 2 = 16 Fourth term
4
an is called the general term.
a5 = 2 5 = 32 Fifth term
an = 2 n
n th term
Practice Problems:
Write the first five terms of the sequence whose general term is given by:
7.
10.
an = 3n -1
an = 2n
8.
an = n 2 + 2
11.
n +1
an =
n
2
9.
an = 3n-2
12.
(-1)n
an =
2n -1
Arithmetic Sequences
Note that in the sequence 2, 5, 8, 11, 14, … 3n -1, … the difference between successive terms is
always 3. The pattern is “add 3” to the previous term to get the next term. Such a sequence is an
arithmetic sequence. These sequences have the following property: The difference between
successive terms is the same constant. This constant is called the common difference. For the
sequence above, the common difference is 3. No matter what term you use when you subtract the
previous term from it the difference will always be 3.
In general, an arithmetic sequence can be defined as follows:
Arithmetic Sequence
Let d be a real number. A sequence an is an arithmetic sequence if
an+1 - an = d
The number d is the common difference for the sequence.
Practice Problems:
Find the common difference of the following arithmetic sequences.
13.
3, 8, 13, 18, …, 5n - 2 , …
14.
11, 7, 3, -1, … -4n +15 , …
15.
2, 4, 6, 8, … 2n, …
16.
-15, -9, -3, 3, …6n – 21, …
3
Let’s look at practice problem 13. The first 5 terms are 3, 8, 13, 18, 23.
Notice the general pattern of the terms:
a1 = 3
a1 = a1
a2 = 3 + 5 = 8
a2 = a1 + d
a3 = 8 + 5 = 13 or 3 + 5 + 5
a3 = a1 + d + d or a1 + 2d
a4 = 13 + 5 = 18 or 3 + 5 + 5 + 5
a4 = a1 + d + d + d or a1 + 3d
a5 = 18 + 5 = 23 or 3 + 5 + 5 + 5 + 5
a5 = a1 + d + d + d + d or a1 + 4d
**Notice the relationship between the term number and the multiplier of d. The multiplier is 1
less than the term number.
Formula for the nth Term of an Arithmetic Sequence:
The nth term of an arithmetic sequence with a common difference of d is
given by
an = a1 + (n -1)d
Practice Problems:
Find the equation for the nth term of an arithmetic sequence whose first term and common
difference is given.
17.
a1 = 4 and d = 3
18.
a1 = 10 and d = -5
Find the equation for the nth term of an arithmetic sequence from the given terms in the
sequence.
19.
2, 7, 12, 17, …
20.
12, 9, 6, 3, …
Find the indicated term of an arithmetic sequence given the first term and common difference.
21.
Find a8 if a1 = 12 and d = 3
22.
4
Find a12 if a1 = 32 and d = -4
Geometric Sequences
We will now investigate a geometric sequence. In the sequence 5, 15, 45, 135, …, each previous
term is multiplied by 3 to get the next term. The pattern of “multiplying by a constant" to get the
next term defines a geometric sequence. The constant is called the common ratio because it is
the ratio of any term (after the first) divided by its preceding term.
135
=3
45
45
=3
15
15
= 3 … 3 is the common ratio
5
Geometric Sequence and Common Ratio r
Let r be a nonzero constant real number. A sequence is a geometric sequence if
an+1
= r for all positive integers n
an
Practice Problems:
Determine if the sequence is a geometric sequence.
23.
2, 8, 32, 128, …
24.
1, 4, 9, 16, …
Let’s look at problem 23. The first four terms are 2, 8, 32, 128.
Notice the general pattern of the terms:
a1 = 2
a1 = a1
a2 = 2 i4
a3 = 8 i4 or 2 i4 i4
a4 = 32 i4 or 2 i4 i4 i4
a2 = a1 ir
or a1 ir1
a3 = a1 ir ir
or a1 ir 2
a4 = a1 ir ir ir or a1 ir 3
**Notice the relationship of the term number and the power of r. The power is 1 less than the
term number.
5
nth Term of a Geometric Sequence
The nth term of a geometric sequence with the first term a1 and common ratio r is
an = a1r n-1
Practice Problems:
Find the equation for the nth term of the geometric sequence.
25.
-4, 12, -36, 108, …
26.
9, 3, 1,
1
,…
3
Find the 7th term of the geometric sequence given the common ratio and the first term of the
sequence.
27.
a1 = 3 and r = 2
28.
6
a1 =
1
1
and r = 2
2