Sequences

Author: John J Weber III, PhD Corresponding Textbook Sections:

• Section 11.1 – Sequences

Expected Educational Results

• Objective 18–01: I can classify a sequence as arithmetic, geometric, or fibonacci.

• Objective 18–02: I can explain what it mean for a sequence to converge and diverge.

• Objective 18–03: I can analytically (i.e., non-graphically and non-numerically) determine if a sequence converges or diverges.

• Objective 18–04: I can analytically (i.e., non-graphically and non-numerically) determine if a sequence is increasing or decreasing.

• Objective 18–05: I can analytically (i.e., non-graphically and non-numerically) determine if a sequence is bounded.

Bloom’s Taxonomy

A modern version of Bloom’s Taxonomy is included here to recognize various different levels of understanding and to encourage you to work towards higher-order understanding (those at the top of the pyramid). All Objectives, Investigations, Activities, etc. are color-coded with the level of understanding.

Sequences

Definition: Sequence

A sequence is a list of numbers written in a specific order. The list is written between $\{$$\left\{\right.$ and $\}$$\left.\right\}$.

Examples of sequences:

1. $\{6,7,8,8,9,1,3,6,9,3\}$$\left\{6, 7, 8, 8, 9, 1, 3, 6, 9, 3\right\}$

2. $\{3,0,0,2,1\}$$\left\{3, 0, 0, 2, 1\right\}$

3. $\{1,2,3,4,5,6,\dots \}$$\left\{1, 2, 3, 4, 5, 6, \ldots\right\}$

4. $\{2,3,5,7,11,13,\dots \}$$\left\{2, 3, 5, 7, 11, 13, \ldots\right\}$

5. $\{5,7,9,11,13,\dots \}$$\left\{5, 7, 9, 11, 13, \ldots\right\}$, $n\ge 3$$n\geq 3$

Definition: Term of a Sequence

One of the numbers in a sequence, a.k.a., element. Terms of a sequence are referenced using the variable $a_n$$a_n$ for $n\ge 0$$n\geq 0$.

In the first sequence above, $(i.)$$(i.)$, the first term of the sequence is $a_0=6$$a_0 = 6$, the second term of the sequence is $a_1=7$$a_1 = 7$, etc.

NOTE: The variable and the starting value are arbitrary, e.g., we could have easily identified the first two terms of the first sequence above, $(i.)$$(i.)$, as $b_1=6$$b_1 = 6$ and $b_2=7$$b_2 = 7$, i.e., $n\ge 1$$n\geq 1$.

Investigation 01

Identify $a_4$$a_4$, for $n\ge 0$$n\geq 0$, in all the sequences above.

Definition: Infinite Sequence

A sequence that can be paired one-to-one with the natural numbers. In other words, there are infinitely-many terms in the sequence.

Investigation 02

Identify which of the above sequences are infinite sequences. Explain.

Definition: $n^{\text{th}}$$n^{\text{th}}$-Term of a Sequence

A general form for any term in a sequence, denoted $a_n$$a_n$, for some starting value for $n$$n$.

For example, the above sequences can be written as:

1. $a_n=$$a_n =$ the digit of Dr. Weber's office phone number, $n\ge 1$$n\geq 1$

2. $b_n=$$b_n =$ the digit of GPC Clarkston's zip code, $n\ge 1$$n\geq 1$

3. $c_n=$$c_n =$ the $n^{\text{th}}$$n^{\text{th}}$ natural (i.e., counting) number, $n\ge 1$$n\geq 1$

4. $d_n=$$d_n =$ the $n^{\text{th}}$$n^{\text{th}}$ prime number, $n\ge 1$$n\geq 1$

Classification of Sequences

Definition: Arithmetic Sequence

A sequence of numbers such that the difference, $d$$d$, between any two consecutive terms is constant. Denoted $a_n=a_0+nd$$a_n=a_0+nd$, $n\ge 0$$n\geq 0$ or $a_n=a_1+(n-1)d$$a_n=a_1+(n-1)d$, $n\ge 1$$n\geq 1$.

Definition: Geometric Sequence

A sequence of numbers such that the common ratio, $r={\displaystyle \frac{a_{n+1}}{a_n}}$$r=\dfrac{a_{n+1}}{a_n}$, of any two consecutive terms is constant. Denoted $a_n=a_0{r}^n$$a_n=a_0r^n$, $n\ge 0$$n\geq 0$ or $a_n=a_1{r}^{n-1}$$a_n=a_1r^{n-1}$, $n\ge 1$$n\geq 1$.

Definition: Fibonacci Sequence

A sequence of numbers defined recursively: $a_0$$a_0$, $a_1$$a_1$, $a_n=a_{n-1}+a_{n-2}$$a_n=a_{n-1}+a_{n-2}$, $n\ge 2$$n\geq 2$.

An alternate way to write the Fibonacci Sequence is explicitly: $F_{n+2}=F_{n+1}+F_n$$F_{n+2}=F_{n+1}+F_{n}$, $n>0$$n>0$, where $F_0$$F_0$ and $F_1$$F_1$ are given.

Interestingly, the explicit formula for the Fibonacci sequence can be written as: $F_{n+1}=F_n+F_{n-1}$$F_{n+1}=F_{n}+F_{n-1}$, $n>1$$n>1$, where $F_0$$F_0$ and $F_1$$F_1$ are given.

Definition: Harmonic Sequence

The sequence: $\{\frac{1}{1},\frac{1}{2},\frac{1}{3},\frac{1}{4},\dots ,\frac{1}{n}\}$$\left\{\frac{1}{1}, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \ldots, \frac{1}{n}\right\}$.

Definition: Alternating Sequence

A sequence where the signs of the terms change,\newline denoted $a_{n+1}=(-1{)}^n{a}_n$$a_{n+1}=(-1)^n a_n$, $n\ge 0$$n\geq 0$ or $a_{n+1}=(-1{)}^{n+1}{a}_n$$a_{n+1}=(-1)^{n+1} a_n$, $n\ge 0$$n\geq 0$.

Method: Classifying Sequences

1. Check for arithmetic sequence:

   1. Compute the following differences

      1. $a_1-a_0$$a_1 - a_0$

      2. $a_2-a_1$$a_2 - a_1$

      3. $a_3-a_2$$a_3 - a_2$

      4. $a_{n+1}-a_n$$a_{n+1} - a_n$

   2. If all the differences are equal, then the sequence has a common difference, i.e., $d$$d$, then

   3. The sequence is an arithmetic sequence represented by $a_n=a_0+nd$$a_n=a_0+nd$, $n\ge 0$$n\geq 0$, where $a_0$$a_0$ is replaced by the first term of the sequence and $d$$d$ is replaced by the common difference.

2. Check for geometric sequence:

   1. Compute the following ratios

      1. ${\displaystyle \frac{a_1}{a_0}}$$\displaystyle \frac{a_1}{a_0}$

      2. ${\displaystyle \frac{a_2}{a_1}}$$\displaystyle \frac{a_2}{a_1}$

      3. ${\displaystyle \frac{a_3}{a_2}}$$\displaystyle \frac{a_3}{a_2}$

      4. ${\displaystyle \frac{a_{n+1}}{a_n}}$$\displaystyle \frac{a_{n+1}}{a_n}$

   2. If all the ratios are equal, then the sequence has a common ratio, i.e., $r$$r$, then

   3. The sequence is a geometric sequence represented by $a_n=a_0{r}^n$$a_n=a_0r^n$, $n\ge 0$$n\geq 0$, where $a_0$$a_0$ is replaced by the first term of the sequence and $r$$r$ is replaced by the common ratio.

3. Check for fibonacci sequence:

   1. Compute the following sums

      1. $a_0+a_1$$a_0+a_1$ and compare to $a_2$$a_2$

      2. $a_1+a_2$$a_1+a_2$ and compare to $a_3$$a_3$

      3. $a_2+a_3$$a_2+a_3$ and compare to $a_4$$a_4$

      4. $a_n+a_{n+1}$$a_n+a_{n+1}$ and compare to $a_{n+2}$$a_{n+2}$

   2. If all the above calculations are equal, then the sequence is a fibonacci sequence.

Investigation 03

Classify the following sequences as arithmetic, geometric, fibonacci, or none of these. Explain.

1. $\{1,2,4,8,16,32,\dots \}$$\left\{1, 2, 4, 8, 16, 32, \ldots\right\}$

2. $\{1,3,5,7,9,11,\dots \}$$\left\{1, 3, 5, 7, 9, 11, \ldots\right\}$

3. $\{-1,1,0,1,1,2,\dots \}$$\left\{-1, 1, 0, 1, 1, 2, \ldots\right\}$

4. $\{1,4,9,16,25,36,\dots \}$$\left\{1, 4, 9, 16, 25, 36, \ldots\right\}$

5. $\{9,3,1,\frac{1}{3},\frac{1}{9},\frac{1}{27},\dots \}$$\left\{9, 3, 1, \frac{1}{3}, \frac{1}{9}, \frac{1}{27}, \ldots\right\}$

Investigation 04

Find the first four terms of the following sequences.

1. ${\displaystyle a_n=\frac{n+1}{n}}$$\displaystyle a_n=\frac{n+1}{n}$, $n\ge 1$$n\geq 1$.

2. ${\displaystyle a_n=(-1{)}^n\frac{3n^2+1}{n+2}}$$\displaystyle a_n=(-1)^n\frac{3n^2+1}{n+2}$, $n\ge 0$$n\geq 0$.

3. ${\displaystyle a_n=n{e}^{-n}}$$\displaystyle a_n=ne^{-n}$, $n\ge 1$$n\geq 1$.

4. ${\displaystyle a_n=(-1{)}^{n+1}\frac{n}{n^2-3n+2}}$$\displaystyle a_n=(-1)^{n+1}\frac{n}{n^2-3n+2}$, $n\ge 3$$n\geq 3$. Explain why $n\ne 0$$n\neq 0$, $n\ne 1$$n\neq 1$, and $n\ne 2$$n\neq 2$.

5. ${\displaystyle a_n=\sqrt{n+1}}$$\displaystyle a_n=\sqrt{n+1}$, $n\ge 0$$n\geq 0$.

Homework

At this time, you should be able to complete the following assignments:

• Section 11.1: # 3, 5, 7, 9, 11, 13, 15, 17, 19, 21.

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Created: Thursday, 5 November 2020 6:42 EDT Last Modified: Tuesday, 14 March 2023 19:25 EDT