# 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 | 0 | 1 | 2 | 3 | \(n\) |
---|---|---|---|---|---|

term | 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).

**Here is a task to try with your student:**

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

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

**Solution: **

- It is possible to describe many patterns in this list.
- 162
- 118,098
- \(2\times 3^n\). This can also be written \(2(3^n)\) or \(2\boldcdot3^n\).
- Here are some ways:
position in list 0 1 2 3 \(n\) term 2 6 18 54 \(2\times 3^n\) “Multiply any term by 3 to get the next term.”

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