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Amy has a master's degree in secondary education and has taught math at a public charter high school.

Watch this video lesson to learn the formula for the general term of an arithmetic sequence. Also learn why this formula makes working with an arithmetic sequence much easier.

An Arithmetic Sequence

We will be talking about the general term of an arithmetic sequence in this video lesson. You will see the formula we use for the general term of any arithmetic sequence and how easy it is to use it.

We begin with our arithmetic sequence, which is a string of numbers where each number is the previous number plus a constant. So, if you look at our number line, you will see that it is an arithmetic sequence since each number is the previous number plus 1.

Number line

This difference between the numbers is called the common difference. We write our sequences inside curly brackets with commas in between the numbers.

Another way of thinking about arithmetic sequences is to think of things that grow over time. For example, we can represent the number of leaves a peach tree has each day with an arithmetic sequence. If our peach tree begins with 10 leaves and grows 15 new leaves each day, we can write the arithmetic sequence {10, 25, 40, . . .} to show the number of leaves that our peach tree has each day. This sequence ends when our peach tree stops growing.

So, while some arithmetic sequences have an end, like our peach tree sequence, some don't and continue on forever, like our number line. Because our arithmetic sequences have a pattern, we have a formula for the general term of an arithmetic sequence.

The General Term

This general term is the formula that is used to calculate any number in an arithmetic sequence.

General term formula

The formula tells us that if we wanted to find a particular number in our sequence, x sub n, we would take our beginning number, a, and add our common difference, d, times n minus 1, which is the location of our desired number minus 1. If we are looking for the 30th number, our n is 30, so our formula begins with x sub 30, and n - 1 equals 29 (30 - 1).

Using the General Term

Let's look at an example to see how we use this general term. Here's a new sequence: {2, 4, 6, . . .}. We want to find the value of the number that is the 25th place in our sequence. Our n in this case is 25. Before we go further, we first have to check whether this sequence is an arithmetic sequence or not.

To do that, we look at the difference between each successive pair of numbers to see if this difference is the same. If it is, then we are looking at an arithmetic sequence. If it isn't, then we can't use our general term formula to find our answer. In our case, we subtract 4 - 2 and also 6 - 4 to see if they equal the same thing. They do; they both equal 2. So, this means the sequence is an arithmetic sequence; it is an arithmetic sequence that jumps by 2 with every number.

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Now that we know this sequence is an arithmetic sequence, now we label our beginning number and our common difference. Our beginning number is 2, so 2 is a. Our common difference is 2, so our d is also 2. Now we go ahead and fill in these numbers into the formula. We get the 25th number is equal to 2 plus 2 times 24. Evaluating this, we get 2 plus 48 equals 50. So, the 25th number is 50.

Why We Use the General Term

Why do we use this general term? Let's go back to our peach tree example. You would use this general term if you, as the peach tree owner, wanted to find out how many leaves the peach tree will have on the 67th day of growing new leaves.

If you use the formula for the general term, you only need to perform three calculations, one for the n minus 1, another to multiply this n minus 1 with the d, and a final one for adding a. Compare doing just 3 calculations to performing 67 additional problems just to find one number. You can see which one is quicker and easier to do.

Lesson Summary

Let's review what we've learned now. An arithmetic sequence is a string of numbers where each number is the previous number plus a constant. This constant difference between each pair of successive numbers in our sequence is called the common difference. The general term is the formula that is used to calculate any number in an arithmetic sequence.

The formula tells us that if we wanted to find a particular number in our sequence, x sub n, we would take our beginning number, a, and add our common difference, d, times n minus 1, which is the location of our desired number minus 1. We use this formula to find a particular number in our sequence. This is particularly useful if our sequence is long, and we want to find a number that is not at the beginning of our sequence.

Learning Outcomes

Following this lesson, you'll have the ability to:

Define arithmetic sequence and common difference

Explain what the general term is and identify a formula for finding it

Describe when you would want to use the formula and how to calculate it

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