Lesson 9
What’s the Equation?
Problem 1
A party will have hexagonal tables placed together with space for one person on each open side:
![A paterrn of hexagons.](https://cms-im.s3.amazonaws.com/DPsVZ1hoACAupMf17HW857Zw?response-content-disposition=inline%3B%20filename%3D%22download.png%22%3B%20filename%2A%3DUTF-8%27%27download.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240630%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240630T171545Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=bd4a100bc60bcad54a30dbffbaaf2acd464034bbb90e65d00210cdca2ce32736)
- Complete this table showing the number of people \(P(n)\) who can sit at \(n\) tables.
\(n\) 1 2 3 4 5 \(P(n)\) 6 - Describe how the number of people who can sit at the tables changes with each step.
- Explain why \(P(3.2)\) does not make sense in this scenario.
- Define \(P\) recursively and for the \(n^{\text{th}}\) term.
Solution
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Problem 2
Diego is making a stack of pennies. He starts with 5 and then adds them 1 at at time. A penny is 1.52 mm thick.
- Complete the table with the height of the stack \(h(n)\), in mm, after \(n\) pennies have been added.
- Does \(h(1.52)\) make sense? Explain how you know.
\(n\) | \(h(n)\) |
---|---|
0 | \(7.6\) |
1 | |
2 | |
3 |
Solution
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Problem 3
A piece of paper has an area of 80 square inches. A person cuts off \(\frac{1}{4}\) of the piece of paper. Then a second person cuts off \(\frac{1}{4}\) of the remaining paper. A third person cuts off \(\frac{1}{4}\) what is left, and so on.
- Complete the table where \(A(n)\) is the area, in square inches, of the remaining paper after the \(n^{\rm th}\) person cuts off their fraction.
- Define \(A\) for the \(n^{\text{th}}\) term.
- What is a reasonable domain for the function \(A\)? Explain how you know.
\(n\) | \(A(n)\) |
---|---|
0 | 80 |
1 | |
2 | |
3 |
Solution
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Problem 4
Here is the recursive definition of a sequence: \(f(1)=35, f(n) = f(n-1) - 8\) for \(n\ge2\).
- List the first 5 terms of the sequence.
-
Graph the value of each term as a function of the term number.
Solution
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(From Unit 1, Lesson 7.)Problem 5
Here is a graph of sequence \(q\). Define \(q\) recursively using function notation.
Solution
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(From Unit 1, Lesson 6.)Problem 6
Here is a recursive definition for a sequence \(f\): \(f(0) = 19, f(n) = f(n-1) - 6\) for \(n \geq 1\). The definition for the \(n^{\text{th}}\) term is \(f(n) = 19 - 6 \boldcdot n\) for \(n\ge0\).
- Explain how you know that these definitions represent the same sequence.
- Select a definition to calculate \(f(20)\), and explain why you chose it.
Solution
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(From Unit 1, Lesson 8.)Problem 7
An arithmetic sequence \(j\) starts 20, 16, . . . Explain how you would calculate the value of the 500th term.
Solution
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(From Unit 1, Lesson 8.)