Introduction

A sequence is a list of numbers.

For example, the following list of numbers is a sequence:

{\color{blue}{4}},\qquad {\color{blue}{8}},\qquad {\color{blue}{12}},\qquad {\color{blue}{16}}

The numbers in a sequence are called terms. For this sequence:

  • the first term is {\color{blue}{4}}

  • the second term is {\color{blue}{8}}

  • the third term is {\color{blue}{12}}

  • the fourth term is {\color{blue}{16}}. This is also called the last term of the sequence.

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Example: Identifying the Terms of a Sequence Given as a List

What is the sum of the first and last terms of the following sequence?

3,\quad-5,\quad7,\quad-9,\quad11,\quad-13,\quad15

EXPLANATION

The first term is 3 and the last term is 15. Their sum is 3+15 = 18.

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Practice: Identifying the Terms of a Sequence Given as a List

What is the fifth term of the following sequence?

\[ 2,\quad -64,\quad -8,\quad 16,\quad 128,\quad 32 \]

a
 $128$
b
 $-8$
c
 $16$
d
 $-64$
e
 $32$
Practice: Identifying the Terms of a Sequence Given as a List

What is the sum of the fourth, fifth, and sixth terms of the following sequence? \[ 8, \quad -24, \quad 44, \quad 92, \quad -244, \quad 458 \]

a
b
c
d
e
Sequences as Functions

Let's return to the following sequence:

{\color{blue}{4}},\qquad {\color{blue}{8}},\qquad {\color{blue}{12}},\qquad {\color{blue}{16}}

For this sequence:

  • the {\color{red}{1}} st term is {\color{blue}{4}}

  • the {\color{red}{2}} nd term is {\color{blue}{8}}

  • the {\color{red}{3}} rd term is {\color{blue}{12}}

  • the {\color{red}{4}} th term is {\color{blue}{16}}

We can represent our sequence using a mapping diagram, as shown below.

The input values refer to the position of each term in the sequence.

Notice that each input value is mapped to one (and only one) output value. Therefore, this sequence is also a function, which we can call f({\color{red}{n}}). For sequences, we usually use the letter n instead of x for the input variable.

So, for this function, we have the following input and output values:

\begin{align*} f({\color{red}{1}}) &= {\color{blue}{4}} \\[5pt] f({\color{red}{2}}) &= {\color{blue}{8}} \\[5pt] f({\color{red}{3}}) &= {\color{blue}{12}} \\[5pt] f({\color{red}{4}}) &= {\color{blue}{16}} \end{align*}

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Sequences With No Last Term

Let's find the sequence generated by the following function:

f({\color{red}{n}}) = {\color{red}{n}} + 1

We can find the first five terms of this sequence by substituting {\color{red}{n}} = {\color{red}{1}},{\color{red}{2}}, {\color{red}{3}},{\color{red}{4}},{\color{red}{5}} into the function.

\begin{align*} f({\color{red}{1}}) &= {\color{red}{1}} + 1 = {\color{blue}{2}} \\[5pt] f({\color{red}{2}}) &= {\color{red}{2}} + 1 = {\color{blue}{3}} \\[5pt] f({\color{red}{3}}) &= {\color{red}{3}} + 1 = {\color{blue}{4}} \\[5pt] f({\color{red}{4}}) &= {\color{red}{4}} + 1 = {\color{blue}{5}} \\[5pt] f({\color{red}{5}}) &= {\color{red}{5}} + 1 = {\color{blue}{6}} \\[5pt] \end{align*}

Therefore, this function generates the following sequence:

{\color{blue}{2}},\qquad {\color{blue}{3}},\qquad {\color{blue}{4}},\qquad {\color{blue}{5}}, \qquad {\color{blue}{6}}, \qquad \ldots

The " \ldots " at the end tells us this sequence continues forever. In other words, this sequence has no last term!

If we want to find, say, the {\color{red}{100}} th term of the sequence, we simply substitute {\color{red}{n}}= {\color{red}{100}} into our function:

f({\color{red}{100}}) = {\color{red}{100}} + 1 = {\color{blue}{101}}

Therefore, the {\color{red}{100}} th term of the sequence is {\color{blue}{101}} .

We usually need to specify which values of {\color{red}{n}} are allowed to be used in our function. To do this, we can write our function as follows:

f({\color{red}{n}}) = {\color{red}{n}} + 1, \qquad {\color{red}{n}} \geq 1

The statement {\color{red}{n}} \geq 1 specifies the function's domain. It tells us we can input integer values greater than or equal to 1 into our function.

Finally, since every term of the sequence can be generated using our function, we say that f({\color{red}{n}}) generates the {\color{red}{n}} th term of the sequence.

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Example: Computing the Terms of a Sequence Using Function Notation

Which of the following statements is true regarding the sequence f(n) =5-2n for n\geq 1?

  1. Its second term is larger than the first term
  2. Its third term is -1
  3. Its last term is -5
EXPLANATION

Let's look at each statement in turn:

  • The first and second terms of the sequence are represented by f(1) and f(2), respectively. We can compute these values using the rule f(n) = 5-2n, as follows; \begin{align} f(1)&=5-2(1)=3\\[5pt] f(2)&=5-2(2)=1 \end{align} We see that the second term f(2)=1 is smaller than the first term f(1)=3. Therefore, statement I is false.

  • We compute the third term as follows: f(3)=5-2(3)=-1 Therefore, statement II is true.

  • The sequence is defined for n\geq 1, which means that there is no limit to how large n can be. There is no last term, because the sequence keeps going on forever. Therefore, statement III is false.

In conclusion, only statement II is true.

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Practice: Computing the Terms of a Sequence Using Function Notation

A sequence is defined as $f(n) =4n-5$ for $n\geq 1.$ The $7$th term of the sequence is

a
b
c
d
e
Practice: Computing the Terms of a Sequence Using Function Notation

Which of the following statements is true regarding the sequence $f(n) =3n^2-1$ for $n\geq 1?$

  1. Its last term is $299$
  2. Its first term is $2$
  3. Its second term is equal to the first term
a
 I only
b
 I and II only
c
 II only
d
 I and III only
e
 II and III only
Practice: Computing the Terms of a Sequence Using Function Notation

A sequence is defined as $f(n) =3n-2$ for $n\geq 1.$ The first three terms of the sequence are the following:

a
b
c
d
e
Sequence Notation

Let's consider the following sequence once more:

{\color{blue}{2}},\qquad {\color{blue}{3}},\qquad {\color{blue}{4}},\qquad {\color{blue}{5}}, \qquad {\color{blue}{6}}, \qquad \ldots

Mathematicians sometimes prefer not to use function notation when describing sequences. The alternative is to use sequence notation. With sequence notation, we refer to the {\color{red}{n}} th term of the sequence as a_{\color{red}{n}}.

So for the sequence listed above, we have:

\begin{align} a_{\color{red}{1}} &= {\color{blue}{2}}\\[5pt] a_{\color{red}{2}} &= {\color{blue}{3}}\\[5pt] a_{\color{red}{3}} &= {\color{blue}{4}}\\[5pt] a_{\color{red}{4}} &={\color{blue}{5}}\\[5pt] a_{\color{red}{5}} &={\color{blue}{6}} \end{align}

Recall that we were able to express this sequence using function notation as follows:

f({\color{red}{n}}) = {\color{red}{n}}+1, \qquad n\geq 1

In sequence notation, the formula looks similar:

a_{\color{red}{n}} = {\color{red}{n}}+1, \qquad n\geq 1

We can calculate the {\color{red}{99}} th term of our sequence by substituting {\color{red}{n}}=99 into the expression a_{\color{red}{n}}{:}

a_{\color{red}{99}} = {\color{red}{99}}+1 = {\color{blue}{100}}

Therefore, the {\color{red}{99}} th term of our sequence is {\color{blue}{100}}.

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Example: Computing the Terms of a Sequence Using Sequence Notation

A sequence is defined as a_n =n^2 for n\geq 1. What is the sum of its second and third terms?

EXPLANATION

Using a_n notation, the second and third terms are a_2 and a_3, respectively.

We can compute these values using the rule a_n=n^2, as follows:

\begin{align} a_2&=2^2=4\\[5pt] a_3&=3^2=9\\[5pt] \end{align}

So, the sum of the second and third terms is

a_2+a_3=4+9=13.

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Practice: Computing the Terms of a Sequence Using Sequence Notation

A sequence is defined as $a_n =2-5n$ for $n\geq 1.$ What is the $4$th term of the sequence?

a
 $-22$
b
 $-18$
c
 $-7$
d
 $-12$
e
 $-15$
Practice: Computing the Terms of a Sequence Using Sequence Notation

A sequence is defined as $a_n =2n+1$ for $n\geq 1.$ What is the sum of the first and third terms?

a
 $12$
b
 $4$
c
 $6$
d
 $8$
e
 $10$
Practice: Computing the Terms of a Sequence Using Sequence Notation

Which of the following statements is true regarding the sequence $a_n =3-n$ for $n\geq 1?$

  1. Its first term is $3$
  2. Its first term is twice the second term
  3. Its last term is $-97$
a
 I only
b
 II only
c
 I and II only
d
 II and III only
e
 I and III only
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