Amy has a master's degree in secondary education and has been teaching math for over 9 years. Amy has worked with students at all levels from those with special needs to those that are gifted.
What Is an Arithmetic Sequence?
What is an arithmetic sequence? It is a string of numbers where each number is the previous number plus a constant, called the common difference. Think of an arithmetic sequence as if you are hosting a candy party where each person that comes is required to bring two candy bars. You are already there with two candy bars. The next person that comes brings two more candy bars. Now you have four candy bars. The next person that comes brings two more candy bars. Now your total is six candy bars. You can see a pattern forming here, and you can probably continue this pattern on your own.
The candy bar example gives you the sequence of 2, 4, 6. We can end there if only two guests came to the party. But we could also continue the pattern indefinitely if we wanted to. We can continue it by adding 2 to 6 to get 8, then adding 2 to 8 to get 10. We would get 2, 4, 6, 8, 10, . . . etc.
Other examples of arithmetic sequences include:
1, 2, 3, 4, . . . etc.
2, 5, 8, 11, . . . etc.
3, 5, 7, 9, . . . and so forth
Notice how all of these sequences have numbers where each number is the previous number plus a constant, a common difference.
Finding the Common Difference
We can calculate the common difference for each of our sequences by taking any two numbers that are next to each other and then subtracting the first from the second. We can repeat with another pair of numbers to make sure that the difference is the same. So for our first sequence of 1, 2, 3, 4, . . ., we can subtract the 1 from the 2 to get 2 - 1 = 1. We can also subtract the 2 from the 3 to get 3 - 2 = 1. Look at that! The common difference is the same - just what we would expect! If we repeated this process with the 3 and the 4, we would see that it also has a difference of 1, so this arithmetic sequence has a common difference of 1.
The second sequence, 2, 5, 8, 11, . . ., has a common difference of 3 because when we subtract the 2 from the 5, we get 3. We also get 3 when we subtract the 5 from the 8 and the 8 from the 11.
Can you find the common difference for the third sequence, 3, 5, 7, 9, etc . . .? If we subtract the 3 from the 5 and the 5 from the 7, we get 2. 7 subtracted from 9 is also 2. So the common difference for this sequence is 2.
Because we have a common difference between all the numbers in our arithmetic sequence, we can use this information to create a formula that allows us to find any number in our sequence, whether it is the 10th number or the 50th number. If you think about it, each number in an arithmetic sequence is actually the first number plus the common difference multiplied by how many times we needed to add it. See, to get to the second term, we added the common difference once to the first term:
To get to the third term, we needed to add the common difference twice. Once to get from the first term to the second, and then once more to get from the second to the third term. If we label a as our first term and d as our common difference, our arithmetic sequence would look like a, a + d, a + 2d, a + 3d, and so on. If n represents the location of a number in this sequence, then the formula to find any number in our sequence is:
Let's go back to our candy bar example to see if this formula really works. Our sequence for the candy bars begins with 2, 4, 6, 8, and so on. Let's check to see if the formula will give us the right term for the fourth term. We already know that it is 8. So let's see if the formula will give us 8 as our answer. We plug in 4 for n since we are looking for the fourth term, 2 for a since our sequence begins with a 2, and 2 for d since 2 is our common difference.
Plugging all this in and solving for x, we get 2 + 2(4 - 1) = 2 + 2(3) = 2 + 6 = 8. Hey, look at that! We get 8! The formula works! So if we wanted to find out how many candy bars we would get if we have 50 people at the party, we can use this formula for n equals 50. Our a is still 2, and our d is also still 2. Plugging in our new n, we have 2 + 2(50 - 1) = 2 + 2(49) = 2 + 98 = 100. We will have 100 candy bars! Awesome!
Now that we are done, and we have a bunch of imaginary candy bars, let's review. An arithmetic sequence is a string of numbers where each number is the previous number plus a constant, called the common difference. To find the common difference we take any pair of successive numbers, and we subtract the first from the second. We should get this same common difference for any other pair of successive numbers in our sequence. The formula to find any number in our sequence is x sub n equals a plus d times n minus 1, where n represents the location of the number in our sequence, a is the first number of our sequence, and d is our common difference.
Following this video lesson, you should be able to:
- Define arithmetic sequence and common difference
- Spot arithmetic sequences
- Describe how to find the common difference
- Recall the formula to find any number in an arithmetic sequence
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