# 13.1 Sequences and their notations

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In this section, you will:
• Write the terms of a sequence defined by an explicit formula.
• Write the terms of a sequence defined by a recursive formula.
• Use factorial notation.

A video game company launches an exciting new advertising campaign. They predict the number of online visits to their website, or hits, will double each day. The model they are using shows 2 hits the first day, 4 hits the second day, 8 hits the third day, and so on. See [link] .

 Day 1 2 3 4 5 … Hits 2 4 8 16 32 …

If their model continues, how many hits will there be at the end of the month? To answer this question, we’ll first need to know how to determine a list of numbers written in a specific order. In this section, we will explore these kinds of ordered lists.

## Writing the terms of a sequence defined by an explicit formula

One way to describe an ordered list of numbers is as a sequence    . A sequence is a function whose domain is a subset of the counting numbers. The sequence established by the number of hits on the website is

$\text{\hspace{0.17em}}\left\{2,4,8,16,32,\dots \right\}.$

The ellipsis (…) indicates that the sequence continues indefinitely. Each number in the sequence is called a term    . The first five terms of this sequence are 2, 4, 8, 16, and 32.

Listing all of the terms for a sequence can be cumbersome. For example, finding the number of hits on the website at the end of the month would require listing out as many as 31 terms. A more efficient way to determine a specific term is by writing a formula to define the sequence.

One type of formula is an explicit formula    , which defines the terms of a sequence using their position in the sequence. Explicit formulas are helpful if we want to find a specific term of a sequence without finding all of the previous terms. We can use the formula to find the $n\text{th}$ term of the sequence , where $n$ is any positive number. In our example, each number in the sequence is double the previous number, so we can use powers of 2 to write a formula for the $n\text{th}$ term.

The first term of the sequence is $\text{\hspace{0.17em}}{2}^{1}=2,\text{\hspace{0.17em}}$ the second term is $\text{\hspace{0.17em}}{2}^{2}=4,\text{\hspace{0.17em}}$ the third term is $\text{\hspace{0.17em}}{2}^{3}=8,\text{\hspace{0.17em}}$ and so on. The $n\text{th}$ term of the sequence can be found by raising 2 to the $n\text{th}$ power. An explicit formula for a sequence is named by a lower case letter $a,b,c...$ with the subscript $n.$ The explicit formula for this sequence is

${a}_{n}={2}^{n}.$

Now that we have a formula for the $n\text{th}$ term of the sequence, we can answer the question posed at the beginning of this section. We were asked to find the number of hits at the end of the month, which we will take to be 31 days. To find the number of hits on the last day of the month, we need to find the 31 st term of the sequence. We will substitute 31 for $n$ in the formula.

If the doubling trend continues, the company will get $\text{2,147,483,648}$ hits on the last day of the month. That is over 2.1 billion hits! The huge number is probably a little unrealistic because it does not take consumer interest and competition into account. It does, however, give the company a starting point from which to consider business decisions.

Another way to represent the sequence is by using a table. The first five terms of the sequence and the $n\text{th}$ term of the sequence are shown in [link] .

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