Unit 1 Family Materials

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 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, . . .

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	0	1	2	3	\(n\)
   term	2	6	18	54	\(2\times 3^n\)

   

   

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

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