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By the end of this lesson, students will be able to:

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By the end of this lesson, students will be able to:

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1. Define and distinguish between arithmetic and geometric sequences
2. Write arithmetic and geometric sequences both recursively and explicitly
3. Translate between recursive and explicit forms of sequences
4. Apply sequence models to real-world situations

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This lesson aligns with A3.FN.7: Write and use arithmetic and geometric sequences recursively and explicitly to model situations, translating between the two forms.

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How can we use sequences to model and predict patterns in real-world situations?

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Guided Practice: Translating Between Forms

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Guided Practice: Translating Between Forms

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In small groups, students will practice translating between recursive and explicit forms of both arithmetic and geometric sequences. Use Padlet to share and discuss solutions.

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Students will individually identify whether given sequences are arithmetic, geometric, or neither, and write them in both recursive and explicit forms.

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Students will research and present real-world examples of arithmetic and geometric sequences (e.g., population growth, compound interest). They'll create Padlet posts to share their findings with the class.

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For students needing additional support, provide a step-by-step guide for identifying and writing sequences.

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Sequences

A sequence $\{ a_{n} \}$ is an infinite list of numbers $$a_{1}, a_{2}, a_{3}, \ldots,$$ where we have one number $a_{n}$ for every positive integer $n$.

Defining sequences.

We can specify a sequence in various ways.

Pattern. We can specify it by listing some elements and implying that the pattern shown continues.

Example.

For example $$2, 4, 6, 8, \ldots$$ would be the sequence consisting of the even positive integers.

Formula. We can also specify a sequence by giving a formula for the term that corresponds to the integer $n$.

Example.

For example the sequence $$2, 4, 6, 8, \ldots$$ can also be specified by the explicit formula $$a_{n} = 2n.$$ Recursively. Finally, we can also provide a rule for producing the next term of a sequence from the previous ones. This is called a recursively defined sequence.

Example.

For example the sequence $$2, 4, 6, 8, \ldots$$ can be specified by the rule $$a_{1} = 2 \quad \text{ and } \quad a_{n} = a_{n-1} +2 \text{ for } n\geq 2.$$ This rule says that we get the next term by taking the previous term and adding $2$. Since we start at the number 2 we get all the even positive integers.

Let's discuss these ways of defining sequences in more detail, and take a look at some examples.

Part 1: Arithmetic Sequences

The sequence we saw in the previous paragraph is an example of what's called an arithmetic sequence: each term is obtained by adding a fixed number to the previous term.

Alternatively, the difference between consecutive terms is always the same.

General Formula.

If a sequence $a_{n}$ is arithmetic, then there is a fixed number $d$ so that $a_{n+1} -a_{n} =d$ for any $n.$ The number $d$ is usually called the step or difference. Let's try to find a formula for the term $a_{n}$ of an arithmetic sequence in terms of $d$ and $a_{1}$.

Let's start with $a_{n} = a_{n-1} + d$. Applying this again, we see that since $a_{n-1} = a_{n-2} + d$ so we get that $a_{n} = a_{n-2} + d +d = a_{n-2} + 2d$. We can continue this way and get: \begin{align*} a_{n} &= a_{n} = a_{n-1} + d \\ &= a_{n-2} + d +d = a_{n-2} + 2d \\ &= a_{n-3} + d + d = a_{n-3} + 3d \\ &\vdots \\ &= a_{2} + (n-2)d \\ &= a_{1} + (n-1)d \\ \end{align*} So we get that in an arithmetic sequence $a_{n}$ with steps of size $d$, the formula for $a_{n}$ is given by: $$a_{n} = a_{1} + (n-1)d$$

Example. Consider the sequence $3, 8, 13, 18, 23, 28, \ldots$. Is it arithmetic? If so, find a formula for $a_{n}$, and use it to find $a_{101}$, the 101st term in the sequence.

Solution. This sequence is arithmetic, since the difference between each term is $5.$
($8-3 = 13-8 = 18-13 = \cdots = 5$.) So this is an arithmetic sequence with step $d=5$ and first term $a_{1} = 3$.
Our formula above gives $a_{n} = a_{1} + (n-1)d = 3 + (n-1)5$.
For $a_{101}$ we plug in $n=101$ into this formula to obtain $a_{101} = 3 + (100)5 = 503$.

Part 2: Geometric Sequences

Consider the sequence $2, 4, 8, 16, 32, 64, \ldots$. This sequence is not arithmetic, since the difference between terms is not always the same. If we look closely, we will see that we obtain the next term in the sequence by multiplying the previous term by the same number. Equivalently, the ratio of consecutive terms is always the same (namely $2$).

A sequence $a_{n}$ where there is a fixed $r$ so that $\frac{a_{n}}{a_{n-1}} = r$ for all $n$ is called a geometric sequence. The number $r$ is usually called the ratio.

General Formula.

Let's try to find the formula for the term $a_{n}$ of a geometric sequence in terms of $r$ and the first term.

Let's start with the relation $\frac{a_{n}}{a_{n-1}} = r$. This gives $a_{n} = r a_{n-1}$. Using this again, we get $a_{n} = r (ra_{n-2}) = r^{2} a_{n-2}$. We can continuer this way and get: \begin{align*} a_{n} &= r a_{n-1} \\ &= r^{2} a_{n-2} \\ &= r^{3} a_{n-3} \\ &\vdots \\ &= r^{n-2} a_{2} \\ &= r^{n-1} a_{1} \\ \end{align*} So we get that for a geometric sequence $a_{n}$ with ratio $r$, the formula for $a_{n}$ is given by: $$a_{n} =r^{n-1} a_{1}$$

Example. Consider the geometric sequence $3, 6, 12, 24, 48, \ldots$. Find a formula for $a_{n}$ and use it to find $a_{7}$.

Solution. To find $r$, we should look at the ratio between successive terms: $r= \frac{a_{1}}{a_{2}} =\frac{6}{3} = 2$. Then using the formula above we get $a_{n} = r^{n-1} a_{1} = 2^{n-1} \cdot 3$.

To find $a_{7}$ we set $n=7$ and get $a_{7} = 2^{7-1} \cdot 3 = 2^{6} \cdot 3 = 64 \cdot 3 = 192$.

Part 3: Recursive Sequences

We have already briefly discussed this idea in the first paragraph. We shall now discuss this in more detail, together with some extra examples.
As we saw in the section on geometric sequences, we can define a geometric sequence either by the rule $a_{n} = r^{n-1} a$, or by the rule that $a_{n} = r a_{n-1}$.
The latter rule is an example of a recursive rule. A recursively defined sequence, is one where the rule for producing the next term in the sequence is written down explicitly in terms of the previous terms.

Let's consider the following (rather famous) example.

Example.

Define a sequence $a_{n}$ as follows: Let $$a_{1} = 1 \quad , \quad a_{2} = 1 \quad \text{, and} \quad a_{n} = a_{n-1} + a_{n-2} \text{ for } n\geq 2.$$ This rule says that to get the next term in the sequence, you should add the previous two terms. Since this rule requires two previous terms, we need to specify the first two terms of the sequence $a_{1}, a_{2}$ to get us started. Using this we can start to list the terms in the sequence, and get $1, 1, 2, 3, 5, 8, 13, 21, 34,\ldots$. (This is the well known Fibonacci sequence.)

Example.

Consider the recursively defined sequence $$a_{1} = 1 \quad , \quad a_{2} = 1 \quad , \quad a_{3} = 1 \quad \text{, and} \quad a_{n} = \frac{a_{n-3}}{a_{n-1} + a_{n-2}} \text{ for } n\geq 3.$$ List the first 7 terms of this sequence.

Solution. \begin{align*} a_{1} &= 1 \\ a_{2} &= 1 \\ a_{3} &= 1 \\ a_{4} &= \frac{1}{1+1} = \frac{1}{2} \\ a_{5} &=\frac{1}{1+\frac{1}{2}} = \frac{1}{\frac{3}{2}} = \frac{2}{3} \\ a_{6} &= \frac{1}{\frac{1}{2} + \frac{2}{3}} = \frac{1}{\frac{7}{6}} = \frac{6}{7} \\ a_{7} &= \frac{ \frac{1}{2}}{\frac{2}{3} + \frac{6}{7}} = \frac{ \frac{1}{2}}{\frac{32}{21}} = \frac{21}{64} \end{align*}

Part 4: Sequences via Lists

The method of using a list to specify a sequence perhaps is the most tricky, since it requires us to look at a short piece of a sequence, and guess at the pattern or rule that is being used to produce the terms in the sequence.
Now that we have seen some more examples of sequences we can discuss how to look for patterns and figure out given a list, how to find the sequence in question.

Example.

When given a list, such as $1, 3, 9, 27, 81, \ldots$ we can try to look for a pattern in a few ways.
Now that we have seen arithmetic, geometric and recursive sequences, one thing we can do is try to check if the given sequence is one of these types.

Arithmetic? To check if a sequence is arithmetic, we check whether or not the difference of consecutive terms is always the same. In this case, the difference changes: $$a_2- a_1 = 3-1 =2 \neq 6 = 9-3 = a_3-a_2.$$ Geometric? To check if a sequence is geometric we check whether or not the ratio of consecutive terms is always the same. In the case it is, so we conclude that the sequence is geometric: $$\frac{3}{1}= \frac{9}{3} = \frac{27}{9} = \frac{81}{27} = 3.$$ This tells us that the sequence is geometric with ratio 3, and initial term 1, so we get that the sequence is given by $$a_{n} = 3^{n-1}.$$ This sequence can also be defined recursively, by the formula $$a_{1} =1 \quad \text{, and} \quad a_{n} = 3a_{n-1} \text{ for } n\geq 2.$$

Example.

Consider the sequence $1, -3, -7, -11, -15, -19, -23, \ldots$. Determine a formula for the $n^{\text{th}}$ term in the sequence.

Solution. We quickly see that this series is not geometric, since $\frac{1}{-3} \neq \frac{-3}{-7}$.

We can now try to see if the sequence is arithmetic. If we look at the differences of consecutive terms, we get: $-3 - 1 = -4 = -7 - (-3) = -11 - (-7)$, so we see that this is an arithmetic sequence with difference $d=-4$. So the general term is $$a_{n} = a_{1} + (n-1) d = 1 +(n-1) (-4) = - 4n + 5.$$
(We could also try to identify a recursive definition of this sequence.)

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A sequence $\{ a_{n} \}$ is an infinite list of numbers $$a_{1}, a_{2}, a_{3}, \ldots,$$ where we have one number $a_{n}$ for every positive integer $n$. We can specify a sequence in various ways. Pattern. We can specify it by listing some elements and implying that the pattern shown continues.

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Challenge advanced students to explore and present on more complex sequences, such as Fibonacci or Lucas sequences.

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End-of-lesson quiz covering identification of sequence types, writing sequences in recursive and explicit forms, and translating between forms.

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Give your students a math quiz that is beautiful, fun and mobile-ready. Customize this template now and let them enjoy problem-solving again!

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Guide a whole-class discussion on the importance of sequences in modeling real-world situations. Ask students to share examples they found most interesting or surprising.

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Students will submit a Padlet post answering: 1) What's the main difference between arithmetic and geometric sequences? 2) How can translating between recursive and explicit forms be useful?

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Complete a worksheet with various sequence problems, including writing sequences in both forms and translating between them.

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Research and create a presentation on a real-world application of either an arithmetic or geometric sequence. Include the sequence in both recursive and explicit forms.

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Play an online game to reinforce understanding of arithmetic and geometric sequences.

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Picture sequencing games offer several benefits for individuals of different ages and cognitive abilities. Here are some of the advantages

Cognitive Development: Picture sequencing games help develop cognitive skills such as logical thinking, problem-solving, and critical reasoning. Players need to analyze the pictures, identify patterns or sequences, and arrange them in the correct order. This process engages their cognitive abilities and enhances their thinking skills.

Memory Enhancement: By arranging a series of pictures in the correct sequence, players need to remember and recall the order of events or steps. This exercise promotes memory enhancement and improves both short-term and long-term memory capabilities.

Language Skills: Picture sequencing games can support language development, especially for younger players. They can learn vocabulary related to different activities or stories depicted in the pictures. Additionally, describing and discussing the events or steps in the sequence can enhance communication and language skills.

Visual-Spatial Skills: These games require players to observe, analyze, and manipulate visual information. By arranging the pictures in the correct order, players enhance their visual-spatial skills, including perception, organization, and spatial awareness.

Sequencing Abilities: Picture sequencing games specifically target the ability to understand and create sequences. This skill is valuable in various areas of life, such as following instructions, organizing tasks, and planning. By practicing sequencing through games, individuals can improve their ability to order events or steps accurately.

Attention and Focus:Engaging with picture sequencing games demands concentration and attention to detail. Players need to observe the pictures carefully, identify the sequence, and arrange them accordingly. Regular practice can help improve attention span and focus.

Problem-Solving and Decision-Making: Sequencing games often involve determining the correct order among several possibilities. This encourages players to evaluate options, make decisions, and solve problems systematically. These problem-solving and decision-making skills are transferable to various real-life situations.

Fun and Engaging:Picture sequencing games provide an interactive and enjoyable learning experience. They are visually appealing and can be designed with engaging themes or stories. This aspect makes learning more enjoyable, especially for younger children, who can develop essential skills while having fun.

Overall, picture sequencing games offer a range of cognitive, memory, language, and problem-solving benefits. They provide an interactive and entertaining way to enhance various skills, making them suitable to promote learning and skill development in an entertaining manner.

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Picture sequencing games offer a range of cognitive, memory, language, and problem-solving benefits. They provide an interactive and entertaining way to enhance various skills, making them suitable to promote learning and skill development in an entertaining manner.

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Explicit formula for sequences

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IXL | Evaluate explicit formulas for sequences | Algebra 2 math

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Improve your math knowledge with free questions in "Evaluate explicit formulas for sequences" and thousands of other math skills.

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Geometric Sequence and Series

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