# Definition and Basic Examples of Arithmetic Sequence

An **arithmetic sequence** is a list of numbers with a **definite pattern**. If you take any number in the sequence then subtract it by the previous one, and the result is always the same or **constant** then it is an arithmetic sequence.

The constant difference in all pairs of *consecutive* or *successive* numbers in a sequence is called the **common difference**, denoted by the letter [latex]d[/latex]. We use the common difference to go from one term to another. How? Take the current term and add the common difference to get to the next term, and so on. That is how the terms in the sequence are generated.

- If the common difference between consecutive terms is positive, we say that the sequence is increasing.
- On the other hand, when the difference is negative we say that the sequence is decreasing.

## Illustrative Examples of Increasing and Decreasing Arithmetic Sequences

Here are two examples of arithmetic sequences. Observe their common differences.

With this basic idea in mind, you can now solve basic arithmetic sequence problems.

### Examples of How to Apply the Concept of Arithmetic Sequence

**Example 1:** Find the next term in the sequence below.

First, find the common difference of each pair of consecutive numbers.

- [latex]15−7 = 8[/latex]

- [latex]23−15 = 8[/latex]

- [latex]31−23 = 8[/latex]

Since the common difference is [latex]8[/latex] or written as [latex]d=8[/latex], we can find the next term after [latex]31[/latex] by adding [latex]8[/latex] to it. Therefore, we have [latex]31 + 8 = 39[/latex].

**Example 2:** Find the next term in the sequence below.

Observe that the sequence is decreasing. We expect to have a common difference that is negative in value.

- [latex]24−31 = −7[/latex]

- [latex]17−24 = −7[/latex]

- [latex]10−17 = −7[/latex]

To get to the next term, we will add this common difference of [latex]d=-7[/latex] to the last term in the sequence. Therefore, [latex]10 + \left( { – 7} \right) = 3[/latex].

**Example 3:** Find the next three terms in the sequence below.

Be careful here. Don’t assume that if the terms in the sequence are all negative numbers, it is a decreasing sequence. Remember, it is decreasing whenever the common difference is negative. So let’s find the common difference by taking each term and subtracting it by the term that comes before it.

The common difference here is positive four [latex]\left( { + \,4} \right)[/latex] which makes this an increasing arithmetic sequence. We can obtain the next three terms by adding the last term by this common difference. Whatever is the result, add again by [latex]4[/latex], and do it one more time.

Here’s the calculation:

The next three terms in the sequence are shown in red.

**Example 4:** Find the seventh term (7^{th}) in the sequence below.

Sometimes you may encounter a problem in an arithmetic sequence that involves fractions. So be ready to use your previous knowledge on how to add or subtract fractions.

Also, always make sure that you understand what the question is asking so that you can have the correct strategy to approach the problem.

In this example, we are asked to find the seventh term, not simply the next term. It is a good practice to write all the terms in the sequence and label them, if possible.

Now we have a clear understanding of how to work this out. Find the common difference, and use this to find the seventh term.

Finding the common difference,

Then we find the 7th term by adding the common difference starting with the 4th term, and so on. Here’s the complete calculation.

Therefore, the seventh term of the sequence is zero (0). We can write the final answer as,

**Example 5:** Find the [latex]\color{red}{35^{th}}[/latex] term in the arithmetic sequence [latex]3[/latex], [latex]9[/latex], [latex]15[/latex], [latex]21[/latex], …

You can solve this problem by listing the successive terms using the common difference. This method is tedious because you will have to keep adding the common difference (which is [latex]6[/latex]) thirty-five times starting with the last term in the sequence.

You don’t have to do this because it is cumbersome. And not only that, it is easy to commit a careless error during the repetitive addition process.

If you decide to find the [latex]\color{red}{35^{th}}[/latex] term of the sequence using this “**successive addition**” method, your solution will look similar below. The “dot dot dot” means that there are calculations there but not shown as it can easily occupy the entire page.

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