# Unit 1 Family Materials

Sequences and Functions

### Sequences and Functions

In this unit, your student will be remembering ways to represent functions. In mathematics, we can think of a function as a rule that tells us how to go from an input to an output. A sequence is a special type of function in which the input is a position in a list, and the output is the number in that position. If you have ever used “fill down” to continue a pattern in a spreadsheet, you have created a sequence. For each sequence of numbers, can you guess a possible rule for creating the next number?

Sequence A: 4, 7, 10, 13, $$\underline{\hspace{.5in}}$$

Sequence B: 2, 6, 18, 54, $$\underline{\hspace{.5in}}$$

You probably noticed that a rule for Sequence A could be “add 3 to any term to get the next term.” There are different ways we could represent this sequence.

Using a table:

 position in list term 0 1 2 3 $$n$$ 4 7 10 13 $$4+3\times n$$

Using a graph:

Using words:

“To find the $$n$$th term, multiply $$n$$ by 3 and add 4.”

Using notation for defining a function:

$$f(n) = 4 + 3 \times n$$ (the value of the $$n$$th term is $$4 + 3 \times n$$). For example, $$f(2) = 4 + 3 \times 2$$, so $$f(2) = 10$$ (the value of the 2nd term is 10).

Let’s revisit Sequence B: 2, 6, 18, 54, . . .

1. Describe any patterns you notice.
2. If the pattern is “multiply any term by 3 to get the next term,” what is the next term?
3. If we call 2 the “0th term,” what is the 10th term?
4. How could we express the $$n$$th term?
5. Represent Sequence B in as many different ways as you can.

Solution:

1. It is possible to describe many patterns in this list.
2. 162
3. 118,098
4. $$2\times 3^n$$. This can also be written $$2(3^n)$$ or $$2\boldcdot3^n$$.
5. Here are some ways:
 position in list term 0 1 2 3 $$n$$ 2 6 18 54 $$2\times 3^n$$

“Multiply any term by 3 to get the next term.”

$$f(n) = 2\times 3^n$$