Representing Sequences: IM Alg2.1.6
[url=https://im.kendallhunt.com/HS/teachers/3/1/6/preparation.html]“Representing Sequences”[/url] from IM Algebra II © 2019 [url=https://www.illustrativemathematics.org/]Illustrative Mathematics[/url]. Licensed under the [url=https://creativecommons.org/licenses/by/4.0/]Creative Commons Attribution 4.0[/url] license.
Table of Contents
- Alg2.1.6 Lesson
- IM Alg2.1.6 Lesson: Representing Sequences
- Alg2.1.6 Practice
- IM Alg2.1.6 Practice: Representing Sequences
IM Alg2.1.6 Lesson: Representing Sequences
For each sequence shown, find either the growth factor or rate of change. Be prepared to explain your reasoning.
5, 15, 25, 35, 45, . . .
Starting at 10, each new term is [math]\frac{5}{2}[/math] less than the previous term.[br]
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[/img]
[math]g(1)=\text{-}5,g(n)=g(n-1)\cdot\text{-}2[/math] for[math]n\ge2[/math]
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[/img]
Take turns with your partner to match a sequence with a recursive definition. It may help to first figure out if the sequence is arithmetic or geometric.
[list][*]For each match that you find, explain to your partner how you know it’s a match.[/*][*]For each match that your partner finds, listen carefully to their explanation. If you disagree, discuss your thinking and work to reach an agreement.[/*][/list][br]There is one sequence and one definition that do not have matches. Create their corresponding match.[br][br]
Complete the table. Drag the Sequences and Definitions to the right place.
Here is a pattern where the number of small squares increases with each new step.
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[/img][br]Write a recursive definition for the total number of small squares [math]S(n)[/math] in Step [math]n[/math].[br]
Sketch a graph of S that shows Steps 1 to 7.
Is this sequence geometric, arithmetic, or neither? Be prepared to explain how you know.
Start with a circle.
[size=150]If you make 1 cut, you have 2 pieces. If you make 2 cuts, you can have a maximum of 4 pieces. If you make 3 cuts, you can have a [i]maximum[/i] of 7 pieces.[/size]
Draw a picture to show how 3 cuts can give 7 pieces.
Find the maximum number of pieces you can get from 4 cuts.
From 5 cuts.
Can you find a function that gives the maximum number of pieces from [math]n[/math] cuts?[br]
IM Alg2.1.6 Practice: Representing Sequences
An arithmetic sequence a starts 2, 5, . . .
Write a recursive definition for this sequence using function notation.[br]
Use your definition to make a table of values for [math]a(n)[/math] and find [math]a(6)[/math].
A geometric sequence g starts 1, 3, . . .
Write a recursive definition for this sequence using function notation.[br]
Sketch a graph of the first 5 terms of g.
Explain how to use the recursive definition to determine [math]g(30)[/math]. (Don’t actually determine the value.)[br]
Match each sequence with one of the recursive definitions. Note that only the part of the definition showing the relationship between the current term and the previous term is given so as not to give away the solutions.
Write the first five terms of each sequence.
[math]a(1)=1,a(n)=3\cdot a(n-1),n\ge2[/math]
[math]b(1)=1,b(n)=\text{-}2+b(n-1),n\ge2[/math]
[math]c(1)=1,c(n)=2\cdot c(n-1)+1,n\ge2[/math]
[math]d(1)=1,d(n)=d(n-1)^2+1,n\ge2[/math]
[math]f(1)=1,f(n)=f(n-1)+2n-2,n\ge2[/math]
[size=150]A sequence has [math]f(1)=120,f(2)=60[/math].[/size][br][br]Determine the next 2 terms if it is an arithmetic sequence, then write a recursive definition that matches the sequence in the form [math]f(1)=120, f(n)=f(n-1)+\underline{\hspace{.25in}}[/math] for [math]n\ge2[/math]
Determine the next 2 terms if it is a geometric sequence, then write a recursive definition that matches the sequence in the form [math]f(1)=120, f(n)=\underline{\hspace{.25in}} \cdot f(n-1)[/math] for [math]n\ge2[/math].
One hour after an antibiotic is administered, a bacteria population is 1,000,000.
Each following hour, it decreases by a factor of [math]\frac{1}{2}[/math].
Complete the table with the bacteria population at the given times.
Do the bacteria populations make a geometric sequence? Explain how you know.