In this unit, students apply their skills using tables, equations, and graphs to identify patterns and learn about sequences. A sequence is a list of numbers, and each number in a sequence is called a term. If you have ever used “fill down” to continue a pattern in a spreadsheet, you have created a sequence. For each sequence of numbers here, can you figure out how to find 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, for sequence \(A\), you can add 3 to any term to get the next term. There are different ways we could represent this sequence.

Using words:

“To find the \(n^{​\text{​th}}\) term, multiply \(n\) by 3 and add 4.”

Using notation for defining a function:

\(A(n) = 4 + 3 \times n\) (the value of the \(n^{​\text{​th}}\) term is \(4 + 3 \times n\)). For example, \(A(2) = 4 + 3 \times 2\), 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. What do you notice about the sequence?
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 \(n^{​\text{​th}}\) term?
5. Represent sequence \(B\) in as many different ways as you can.

Solution:

1. Students may notice:
   1. All the numbers are even.
   2. The values of the terms are increasing.
   3. Each term is 3 times the value of the previous term.
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.”

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