# 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:

1. Complete this table showing the number of people $$P(n)$$ who can sit at $$n$$ tables.
 $$n$$ $$P(n)$$ 1 2 3 4 5 6
2. Describe how the number of people who can sit at the tables changes with each step.
3. Explain why $$P(3.2)$$ does not make sense in this scenario.
4. Define $$P$$ recursively and for the $$n^{\text{th}}$$ term.

### 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.

1. Complete the table with the height of the stack $$h(n)$$, in mm, after $$n$$ pennies have been added.
2. Does $$h(1.52)$$ make sense? Explain how you know.
$$n$$ $$h(n)$$
0 $$7.6$$
1
2
3

### 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.

1. 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.
2. Define $$A$$ for the $$n^{\text{th}}$$ term.
3. What is a reasonable domain for the function $$A$$? Explain how you know.
$$n$$ $$A(n)$$
0 80
1
2
3

### Problem 4

Here is the recursive definition of a sequence: $$f(1)=35, f(n) = f(n-1) - 8$$ for $$n\ge2$$.

1. List the first 5 terms of the sequence.
2. Graph the value of each term as a function of the term number.

### Solution

(From Unit 1, Lesson 7.)

### Problem 5

Here is a graph of sequence $$q$$. Define $$q$$ recursively using function notation.

### Solution

(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 $$n\ge0$$.

1. Explain how you know that these definitions represent the same sequence.
2. Select a definition to calculate $$f(20)$$, and explain why you chose it.

### Solution

An arithmetic sequence $$j$$ starts 20, 16, . . . Explain how you would calculate the value of the 500th term.