Student Library

Sequences

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What is a Sequence?

Objectives

  • Define what is a sequence.
  • Describe the similarities between a function and a sequence.
  • Describe the terms of an explicitly defined sequence.
  • Describe the terms of a recursively defined sequence.

Definition of a Sequence

Simply said, a sequence is an ordered set of numbers.
For instance, these are some examples of sequences:

Misplaced &\align{ &\brc{1, 2, 3, 4, 5, 6, 7, 8, 9, 10}\\[1ex] &\brc{1, \frac{1}2, \frac{1}4, \frac{1}8, \frac{1}{16}, \frac{1}{32}}\\[1ex] &\brc{1, 2, 1, -2, -7, -14, -23}\\[1ex] &\brc{2, 2, 4, 6, 10, 16, 26, 42} }

We can refer to a specific sequence with a capital letter.

Misplaced &\align{ A&=\brc{1, 2, 3, 4, 5, 6, 7, 8, 9, 10}\\[1ex] B&=\brc{1, \frac{1}2, \frac{1}4, \frac{1}8, \frac{1}{16}, \frac{1}{32}}\\[1ex] C&=\brc{1, 2, 1, -2, -7, -14, -23}\\[1ex] D&=\brc{2, 2, 4, 6, 10, 16, 26, 42} }

We refer to the items inside the sequence as terms of a sequence.
Since a sequence is an ordered set, we can refer to a specific term by referring to its position in the sequence.
We call the position of a term in a sequence as its index.

Misplaced &\array{c}{A = \brc{1, 2, 3, 4, 5, 6, 7, 8, 9, 10}\\\\ \array{cc}{ \text{1st term of }A = 1 & \text{2nd term of }A = 2\\ \text{3rd term of }A = 3 & \text{4th term of }A = 4\\ }}

Referring to specific terms like this can become clunky.
Instead, we use the notation to refer to the first term of the sequence .

Misplaced &\array{c}{A = \brc{1, 2, 3, 4, 5, 6, 7, 8, 9, 10}\\[1ex] \array{cc}{ a_1= 1 & a_2 = 2\\ a_3 = 3 & a_4 = 4\\ }}

We can use this notation to refer terms in other sequences, such as in sequence , by using its small letter counterpart, .

Worked Example #1
Referring and Locating Specific Terms in a Sequence

Let , , and be sequences where:

Misplaced &B &= \brc{\frac{1}9, \frac{1}3, 1, 3, 9, 27, 81}\\ C &= \brc{6, 1, 7, 8, 15, 23, 38, 61, 99, 160}}$$ Use the notation $a_n$ to refer to these specific terms: 1. $29$ in sequence $A$ 2. $\dfrac{1}3$ in sequence $B$ 3. $38$ in sequence $C$ 4. $9$ in sequence $B$ 5. $-1$ in sequence $A$ Locate the terms expressed in the $a_n$ notation. 1. $a_8$ 2. $b_5$ 3. $b_3$ 4. $c_2$ 5. Evaluate $c_{10}-b_7$ >[!solution]- >For the first problem: >1. In sequence $A$, $29$ is the $10^{\text{th}}$ term so $a_{10} = 29$. >2. In sequence $B$, $\dfrac{1}3$ is the $2^{\text{nd}}$ term so $b_{2} = \dfrac{1}3$. >3. In sequence $C$, $38$ is the $7^{\text{th}}$ term so $c_{7} = 38$. >4. In sequence $B$, $9$ is the $5^{\text{th}}$ term so $b_{5} = 9$. >5. In sequence $A$, $-1$ is the $4^{\text{th}}$ term so $a_{4} = -1$ > >For the second problem: >1. $a_8$ refers to the $8^{\text{th}}$ term in sequence $A$ and that is $19$. >2. $b_5$ refers to the $5^{\text{th}}$ term in sequence $B$ and that is $9$. >3. $b_3$ refers to the $3^{\text{rd}}$ term in sequence $B$ and that is $1$. >4. $c_2$ refers to the $2^{\text{nd}}$ term in sequence $BC$ and that is also $1$. >5. $c_{10}$ is $160$, while $b_7$ is $81$. Therefore, the expression $c_{10}-b_7$ evaluates to $160 - 81 = 79$.

Therefore, using these information, we can define a sequence as:

Definition
Sequence

A sequence is an ordered set of numbers in the form:

where are the terms of the sequence .
The notation represents the term of sequence , where is the index or the position of the term in the sequence

Most often, we can define sequences by simply listing the individual terms.

Defining sequences this way is called the roster method.
However, this becomes very inconvenient if we want more terms in our sequence

Explicit Sequences

Sequences can be defined in various ways.
One way we can define them is by using a function to explicitly define a sequence.
For example, if we have a sequence:

We can automatically generate the first 5 terms of this sequence just from this rule here:

Misplaced &\align{ a_1 &= 1^2+1 = 2\\ a_2 &= 2^2+1 = 5\\ a_3 &= 3^2+1 = 10\\ a_4 &= 4^2+1 = 17\\ a_5 &= 5^2+1 = 26\\ }

Sequences defined from an explicit function relation are called explicit sequences.

Definition
Explicit Sequences

An explicit sequence is a sequence where each term of the sequence is defined as:

where the function is defined for all .

Our definition notes an interesting difference of sequences from functions.
While a function may be defined for all real numbers , an explicit sequence is only be defined for the natural numbers only.

This is because represents the position of a specific term or the order at which it appears in a sequence. Thus, it would not make sense to have term or a term in a sequence.

Worked Example #2
Finding Terms of an Explicitly-Defined Sequence

Find the first terms of the following sequences.

Recursive Sequences

Aside from defining sequences as an explicit function definition, we can also define a sequence in a different way.

We can define a sequence by making rules on how to get the next term as a function of its previous terms. To see how it looks like, let us take an example.

This sequence tells us that to get the next term in our sequence, we take the previous term , then substitute it on our expression .

However, this isn’t enough to build our sequence. We need a “starting position” for us to build our sequence.

We can then establish that , then by using the rule from before, we can begin building our sequence.

Misplaced &\array{c}{ \system{&a_1 = 2\\ &a_n = 2+5a_{n-1}}\\[3ex] \array{l}{ a_1 = 2\\ a_2 = 2+5a_1 = 2+5(2) = 12\\ a_3 = 2+5a_2 = 2+5(12) = 62\\ a_4 = 2 + 5a_3 = 2+5(62) = 312\\ a_5 = 2 + 5a_4 = 2+5(312) = 1562 } }

Here, we establish a sequence where the next term is based on its previous terms.
However, to build our sequence, we define a single term in our sequence. This term is what we call the base case of a recursive sequence.

Definition
Recursive Sequences

A recursive sequence is a sequence where the next successive term is defined as a function of its preceding terms .

We then define a recursive sequence by defining any term in the sequence that serves an initial term called a base case, then using the rules to get the next succeeding term.

In general, a recursive sequence has the form:

Misplaced &\system{&a_1 = c\\ &a_n = f(a_{n-1})}

To familiarize ourselves to the concept of a recursive sequence, we dive into it by working our some practice problems:

Worked Example #3
Finding a Specific Term in a Recursive Sequence

Find if the sequence is defined as:
Misplaced &\system{&a_1 = -9\\&a_n = -\frac{1}3a_{n-1}}

As we can see, we can define sequences recursively to create unique sequences that are otherwise impossible with explicit function definitions.

One example of these sequences is the Fibonacci Sequence.
The Fibonacci sequence is a unique kind of sequence that’s mostly associated, not only with patterns in nature, but with designs in art, architecture, and even the universe itself.

To construct a Fibonacci sequence, we define it as the sum of the previous two terms.

Definition
Fibonacci Sequence

The Fibonacci sequence is a special sequence that is defined as the sum of the last two preceding terms. Mathematically, the sequence can be defined recursively as:

Misplaced &&f_0 = 0\\ &f_1 = 1\\ &f_n = f_{n-2}+f_{n-1} }$$

Worked Example #4:
Finding Terms on a Fibonacci Sequence

Use the definition of the Fibonacci sequence to find its term.

Arithmetic Sequences

Objectives

  • Define what is an arithmetic sequence.
  • Identify an arithmetic sequence.
  • Describe an arithmetic sequence defined explicitly and recursively.
  • Find terms on an arithmetic sequence defined explicitly and recursively.
  • Deduce the common difference given an arithmetic sequence.

In this and the following sections, we will explore some types of sequences that often had many applications in the real world. We can begin tackling them by observing this particular sequence below:

One good observation is when we plot these points in a graph.
Here, the -axis represents the index of our sequence, and the -axis for the value.

{insert photo}

As we can see, the plot of points resemble a straight line.
However, we can see more connections as we take the difference between each consecutive term.

Misplaced &a_2 -a_1 &= 7 -3 = 4\\ a_3 - a_2 &= 11 - 7 = 4\\ a_4 - a_3 &= 15 -11 = 4\\ a_5 - a_4 &= 19 -15 = 4\\ a_6 - a_5 &= 23 - 19 = 4\\ a_7 - a_6 &= 27 - 23 = 4\\ a_8 - a_7 &=31 - 23 = 4\\ }$$ One good characteristic of these sequences is that each consecutive term has a **common difference**. This also means that we can get the next term by adding the same constant. In fact, sequences that exhibit this characteristic are called **arithmetic sequences**. > [!definition] **Definition<br>Arithmetic Sequences** > > An arithmetic sequence is a sequence where the difference between each consecutive term is a constant value called the **common difference**. >[!problem] **Worked Example \#5 <br>Identifying Arithmetic Sequences** > >Identify if the following sequences are arithmetic or not. >1. $A = \brc{-3, 4, 11, 19, 25, 33, 40, 47}$ >2. $B = \brc{8, 5, 2, -1, -4, -7, -10, -13}$ >3. $C = \brc{\frac{1}{16}, \frac{1}{8}, \frac{1}{4}, \frac{1}{2}, 1, 2, 4, 8}$ >4. $d_n = 2(n-1)$ >5. $\dstyle\system{&g_1 = 1\\ &g_n=3g_{n-1}}$ > >>[!solution]- >>To determine that each sequence is an arithmetic sequence, we identify if each consecutive term has a common difference. >>1. $A$ has a difference of $7$ on most consecutive terms but between $a_3$ and $a_4$, the difference is $8$, therefore this is not an arithmetic sequence. >>2. $B$ has a difference of $-3$ on all consecutive terms. This is still an arithmetic sequence as the difference between each term is constant. >>3. $C$ is not an arithmetic sequence because the difference of each consecutive term is not constant: $a_6-a_5 = 1$ while $a_7 - a_6 = 2$. >>4. Sequence $D$ is defined explicitly. Getting its first $5$ terms, we get the sequence: $D = \brc{2, 4, 6, 8, 10}$. Each term has a common difference of $2$, therefore this is an arithmetic sequence. >>5. Sequence $G$ is defined recursively. By getting its first $5$ terms, we get the sequence $G = \brc{1, 3, 6, 9, 27}$. This is not an arithmetic sequence as the difference between each consecutive term is not constant. Now that we get a feel on what an arithmetic sequence looks like, we can generalize it for any possible arithmetic sequence. We can start from an arbitrary sequence with the terms $a_1, a_2, a_3, \ldots$ up until $a_n$. $$\brc{a_1, a_2, a_3, a_4, a_5, a_6, \ldots,a_{n-1}, a_n}$$ We will assume that this sequence is an arithmetic sequence. Therefore, to get the next term, we add the common difference $d$ from the previous term. $$\brc{a_1, a_1+d, a_2 +d, a_3+d, a_4+d , \ldots}$$ Since $a_2 = a_1 +d$ and $a_3 = a_2+d$, and so on, we can use this relation to simplify the expression on each term. $$\align{ &\brc{a_1,\; \par{a_1+d},\; \par{a_1+d+d},\; \par{a_1+d+d+d},\; \par{a_1+d+d+d+d},\;\ldots}\\ &=\brc{a_1,\; a_1+d,\; a_1+2d,\; a_1+3d,\; a_1+4d,\;a_1+5d,\; a_1+6d,\; a_1+7d,\;\ldots} }

This shows us that to get any term in an arithmetic sequence, we add a multiple of to our first term, . The multiple of must be less than the index of , therefore we can define any arithmetic sequence explicitly as:

Definition
General Explicit Form of an Arithmetic Sequence

Every arithmetic sequence, when defined explicitly, can be written as:

where:
is the term of the sequence
is the term of the sequence
is the common difference
is the index/position of

Geometric Sequences

Objectives

  • Define what is a geometric sequence.
  • Describe a geometric sequence defined explicitly and recursively.
  • Find terms on an arithmetic sequence defined explicitly and recursively.
  • Deduce the common ratio given a geometric sequence

Graph View