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10.6 Sequences
A sequence is a function with domain a set of positive, successive integers. Let’s
consider the function f (n ) = 2n − 1 . The domain is the set of natural numbers. We can
consider rewriting the function as a n = 2n − 1 . The elements in the range are called
terms of the sequence: a1 is the first term, a 2 the second term, and a n the nth term, or
the general term:
a1 = 2(1) – 1 = 1
a 2 = 2(2) – 1 = 3
a3 = 2(3) – 1 = 5
.
.
.
The ordered list of elements 1, 3, 5, …, 2n -1, … is often informally referred to as a
sequence. A sequence can also be represented in the abbreviated form {a n }, where the
symbol for the nth term is placed between braces. We refer to the sequence as the
sequence {2n – 1}. The sequence can either be a finite sequence, where we know how
many terms we want to list, or it can be an infinite sequence, where the last term is the nth
term. The sequence above is an example of an infinite sequence, because we are not told
how many terms to list.
Example
Write the first four terms for the sequence a n =
1+1 2
=
3
3
2 +1 3
= =1
a2 =
3
3
3+1 4
=
a3 =
3
3
4 +1 5
=
a4 =
3
3
n +1
.
3
a1 =
 n + 1
The first four terms in the sequence 
 are 2/3, 1, 4/3, and 5/3.
 3 
Sometimes a function is not given for us to determine its sequence. This type of formula
is called a recursive formula. A recursive formula allows us to find a number given its
preceding number. Let’s consider the recursive formula
a1 = 2
a n = an −1 + 3 , for n ≥ 2.
Notice that we are unable to determine the fourth number without knowing the third.
And, we are unable to determine the third number without knowing the second. To list
the terms of the sequence:
a1 = 2
a 2 = a 2−1 + 3 = a1 + 3 = 2 + 3 = 5
a3 = a3−1 + 3 = a 2 + 3 = 5 + 3 = 8
a 4 = a 4−1 + 3 = a3 + 3 = 8 + 3 = 11
.
.
.
a n = an −1 + 3
Example
Write the first five terms of the sequence a1 = 4, a n = 2a n −1 − 7 , for n ≥ 2.
a1 = 4
a 2 = 2a 2−1 − 7 = 2a1 − 7 = 8 − 7 = 1
a3 = 2a3−1 − 7 = 2a2 − 7 = 2 − 7 = −5
a 4 = 2a 4−1 − 7 = 2a3 − 7 = −10 − 7 = −17
a5 = 2a5−1 − 7 = 2a 4 − 7 = −34 − 7 = −41
Finding a general term of a sequence given the first n terms can be determined by
examining the sequence and deducing a formula that produces the same sequence.
Let’s consider the first four terms in the sequence 2, -4, 8, -16,… Each term can be
written as the product of a power of 2 and a power of -1.
2 = (-1)0(2)1
-4 = (-1)1(2)2
8 = (-1)2(2)3
-16 = (-1)3(2)4
If we choose the domain to be all natural numbers, then a solution is
an = (-1)n-1(2)n
Example
Find the general term of a sequence whose first four terms are 5, 8, 11, 14,…
Notice that each term is three more than the previous term. This means that the nth term
is 3n added to some number. Let’s examine how each term is determined.
5 = 3(1) + 2
8 = 3(2) + 2
11 = 3(3) + 2
14 = 3(4) + 2
In turn, a formula for the sequence can be an = 3n + 2.
If a1, a2, a3, …, an, … is a sequence, then the expression a1 + a2 + a3 + … + an + … is
called a series. If the sequence is finite, the corresponding series is a finite series. If the
sequence is infinite, the corresponding series is an infinite series. We will only be
concerned with finite series for this lesson.
Let’s consider writing the following series using summation notation:
1−
1 1 1 1 1
+ − + −
2 3 4 5 6
What does it mean to use summation notation? We want to create a formula that
produces the numbers that are being added and subtracted. Let’s peel apart the series and
1 1
1 1
1
write it as a sequence: 1, − , , − , , − . Now let’s figure out how each number
2 3 4 5
6
can be determined by using a formula.
1
a1 = 1 = ( −1) 0
1
1
a2 = – ½ = ( −1)1
2
1
a3 = 1 = ( −1) 2
3
3
1
a4 = – ¼ = ( −1) 3
4
1
a5 = 1 = ( −1) 4
5
5
1
a6 = − 1 = ( −1) 5
6
6
1
We can deduce the formula that produces each term to be an = ( −1) n −1
n
Since we are determining the sum of the first six terms, we want to begin with n = 1 and
end with n = 6.
We will be using the symbol ∑ to denote summation notation. The variable n can be
replaced by any other variable. Typically, you will see n and k being used in summation
notation.
For this problem we see that we are taking the sum of the first six numbers in a sequence.
Therefore, the summation notation for this series is
6
1
( −1) n −1
∑
n
n =1
Example
Write the series 1 + 2 + 4 + 8 + 16 using summation notation with the summing index k
starting at k = 1.
Let’s first figure out how each number is determined.
1 = 2 0 = 21−1
2 = 21 = 2 2−1
4 = 2 2 = 2 3−1
8 = 2 3 = 2 4−1
16 = 2 4 = 2 5−1
We can deduce the formula that produces each term to be an = 2 n −1
Since we are determining the sum of the first five terms, we want to begin with k = 1 and
end with k = 5.
Therefore, the summation notation for this series is
5
∑2
k −1
k =1
Try the following:
1. Write the first four terms for the sequence a n =
2n − 1
.
n +1
2. Write the first five terms of the sequence a1 = -3, a n = 5a n −1 + 1 , for n ≥ 2.
3. Find the general term of a sequence whose first four terms are 3, 5, 7, 9,…
4. Write the series 2 + 3 + 4 + 5 + 6 using summation notation with the summing index k
starting at k = 1.
Answers
1. ½ , 1, 5/4, 7/5
2. -3, -14, -69, -344, -1719
3. a n = 2n + 1
4.
5
∑k +1
k =1