Halley's Comet appears in the sky approximately every 76 years. The comet was first spotted in the year 1531. Find the

### Arithmetic Sequence

In this concept we will begin looking at a specific type of sequence called an **arithmetic sequence**. In an arithmetic sequence the difference between any two consecutive terms is constant. This constant difference is called the **common difference**. For example, question one in the Review Queue above is an arithmetic sequence. The difference between the first and second terms is

Since the same value, the common difference,

Given the sequence:

Now we can generalize this into a rule for the

#### Solve the following problems

Find the common difference and

To find the common difference we subtract consecutive terms.

Now we can put our first term and common difference into the

Find the

We have what we need to plug into the rule:

Now to find the

Find the

This one is a little less straightforward as we will have to first determine the first term from the term we are given. To do this, we will replace

Now that we have the first term and the common difference we can follow the same process used in the previous example to complete the problem.

Now we can find the

### Examples

#### Example 1

From the information given, we can conclude that

We now have what we need to plug into the rule:

Now to find the

#### Example 2

Find the common difference and the

The common difference is

#### Example 3

Write the \begin{align*}n^{th}\end{align*} term rule and find the \begin{align*}45^{th}\end{align*} term for the arithmetic sequence with \begin{align*}a_{10}=1\end{align*} and \begin{align*}d=-6\end{align*}.

To find the first term:

\begin{align*}a_1+(10-1)(-6)&=1 \\ a_1-54&=1 \\ a_1&=55\end{align*}

Find the \begin{align*}n^{th}\end{align*} term rule: \begin{align*}a_n=55+(n-1)(-6)=55-6n+6=-6n+61\end{align*}.

Finally, the \begin{align*}45^{th}\end{align*} term: \begin{align*}a_{45}=-6(45)+61=-209\end{align*}.

#### Example 4

Find the \begin{align*}62^{nd}\end{align*} term for the arithmetic sequence with \begin{align*}a_1=-7\end{align*} and \begin{align*}d=\frac{3}{2}\end{align*}.

This time we will not simplify the \begin{align*}n^{th}\end{align*} term rule, we will just use the formula to find the \begin{align*}62^{rd}\end{align*} term: \begin{align*}a_{62}=-7+(62-1) \left(\frac{3}{2}\right)=-7+61 \left(\frac{3}{2}\right)=- \frac{14}{2}+ \frac{183}{2}= \frac{169}{2}\end{align*}.

### Review

Identify which of the following sequences is arithmetic. If the sequence is arithmetic find the \begin{align*}n^{th}\end{align*} term rule.

- \begin{align*}2, 3, 4, 5, \ldots\end{align*}
- \begin{align*}6, 2, -1, -3, \ldots\end{align*}
- \begin{align*}5, 0, -5, -10, \ldots\end{align*}
- \begin{align*}1, 2, 4, 8, \ldots\end{align*}
- \begin{align*}0, 3, 6, 9, \ldots\end{align*}
- \begin{align*}13, 12, 11, 10, \ldots\end{align*}
- \begin{align*}4, -3, 2, -1, \ldots\end{align*}
- \begin{align*}a, a+2, a+4, a+6, \ldots\end{align*}

Write the \begin{align*}n^{th}\end{align*} term rule for each arithmetic sequence with the given term and common difference.

- \begin{align*}a_1=15\end{align*} and \begin{align*}d=-8\end{align*}
- \begin{align*}a_1=-10\end{align*} and \begin{align*}d= \frac{1}{2}\end{align*}
- \begin{align*}a_3=24\end{align*} and \begin{align*}d=-2\end{align*}
- \begin{align*}a_5=-3\end{align*} and \begin{align*}d=3\end{align*}
- \begin{align*}a_{10}=-15\end{align*} and \begin{align*}d=-11\end{align*}
- \begin{align*}a_7=32\end{align*} and \begin{align*}d=7\end{align*}
- \begin{align*}a_{n-2}=3n+2\end{align*}, find \begin{align*}a_n\end{align*}

### Answers for Review Problems

To see the Review answers, open this PDF file and look for section 11.5.