How to find the common difference of an arithmetic sequence given two terms

Video Transcript

Find the common difference of the arithmetic sequence given the general term 𝑎 sub 𝑛 is equal to negative three 𝑛 minus one.

Remember, an arithmetic sequence is one in which the difference between consecutive terms is constant. And it’s this common difference that we’re asked to find in this question. There are two ways that we could do this. The first method is we could use the formula for the general term 𝑎 sub 𝑛 equals negative three 𝑛 minus one to calculate some of the terms in this sequence and then look at what the common difference is.

For example, for the first term in the sequence, 𝑛 is equal to one. So substituting, we have 𝑎 sub one is equal to negative three multiplied by one minus one. That’s negative three minus one, which is negative four. For the second term, 𝑛 is equal to two. So we have 𝑎 sub two is equal to negative three multiplied by two minus one. That’s negative six minus one, which is equal to negative seven. And to be on the safe side, let’s calculate one more. For the third term, 𝑛 is equal to three. So we have negative three multiplied by three minus one. That’s negative nine minus one, which is negative 10.

We’ve, therefore, calculated the first three terms in this sequence. And indeed, we could’ve calculated any successive terms by substituting any values of 𝑛. Let’s now look at these three values to determine the common difference. To get from negative four to negative seven, we have to subtract three. And to get from negative seven to negative 10, we also have to subtract three. This tells us that to get from one term to the next throughout our arithmetic sequence, we subtract three. And therefore, the common difference is negative three.

The other way we could answer this question without calculating any terms is to use the general term we’ve been given. And this requires a little bit more familiarity with arithmetic sequences. The general term of an arithmetic sequence can always be expressed in the form 𝑏𝑛 plus 𝑐. That’s some multiple of 𝑛 plus a constant. It’s always the case that the coefficient of 𝑛 in the general term gives the common difference of the sequence. The general term for our sequence is negative three 𝑛 minus one. And so we see that the coefficient of 𝑛 is negative three.

Using two different methods then, we’ve found that the common difference of the arithmetic sequence with a general term negative three 𝑛 minus one is negative three.

We have a new and improved read on this topic. Click here to view

We have moved all content for this concept to for better organization. Please update your bookmarks accordingly.

To better organize out content, we have unpublished this concept. This page will be removed in future.

Finding the nth Term Given Two Terms for an Arithmetic Sequence

Identify common difference, use it to state a general rule

Practice

• Preview
• Assign Practice

Preview

Calculus Sequences and Series in Calculus

• More
  ()

• All Modalities
• Share with Classes

Finding the nth Term Given Two Terms

Loading... 

---

Found a content error?
Tell us

Notes/Highlights

Color	Highlighted Text	Notes
Show More

Image Attributions

ShowHide Details

All Algebra 1 Resources

Which of the following cannot be three consecutive terms of an arithmetic sequence?

Correct answer:

Explanation:

In each group of numbers, compare the difference of the second and first terms to that of the third and second terms. The group in which they are unequal is the correct choice.

The last group of numbers is the correct choice.

Consider the arithmetic sequence

. 

If , find the common difference between consecutive terms.

Correct answer:

Explanation:

In arithmetic sequences, the common difference is simply the value that is added to each term to produce the next term of the sequence. When solving this equation, one approach involves substituting 5 for  to find the numbers that make up this sequence. For example, 

so 14 is the first term of the sequence. However, a much easier approach involves only the last two terms,  and .

The difference between these expressions is 8, so this must be the common difference between consecutive terms in the sequence.

Find the common difference in the following arithmetic sequence.

Correct answer:

Explanation:

An arithmetic sequence adds or subtracts a fixed amount (the common difference) to get the next term in the sequence. If you know you have an arithmetic sequence, subtract the first term from the second term to find the common difference.

Find the common difference in the following arithmetic sequence.

Correct answer:

Explanation:

An arithmetic sequence adds or subtracts a fixed amount (the common difference) to get the next term in the sequence. If you know you have an arithmetic sequence, subtract the first term from the second term to find the common difference.

(i.e. the sequence advances by subtracting 27)

What is the common difference in this sequence?

Correct answer:

Explanation:

The common difference is the distance between each number in the sequence. Notice that each number is 3 away from the previous number.

What is the common difference in the following sequence?

Correct answer:

Explanation:

What is the common difference in the following sequence?

Common differences are associated with arithematic sequences. 

A common difference is the difference between consecutive numbers in an arithematic sequence. To find it, simply subtract the first term from the second term, or the second from the third, or so on...

See how each time we are adding 8 to get to the next term? This means our common difference is 8.

What is the common difference in the following sequence:  

Correct answer:

Explanation:

The common difference in this set is the linear amount spaced between each number in the set.

Subtract the first number from the second number.

Check this number by subtracting the second number from the third number.

Each spacing, or common difference is:  

What is the common difference?  

Correct answer:

Explanation:

The common difference can be determined by subtracting the first term with the second term, second term with the third term, and so forth.   The common difference must be similar between each term.

The distance between the first and second term is .

The distance between the second and third term is .

The distance between the third and fourth term is .

The fractions may seem as though they have a common difference since the denominators are increasing by one for each term, but there is no common difference among the numbers.

The answer is:  

What is the common difference in the following set of data?   

Correct answer:

Explanation:

In order to determine the common difference, subtract the first term from the second term.

Verify that this is the same for the difference of the third and second terms.

The set of data is increasing at increments of five.  

The common difference is:  

All Algebra 1 Resources

Can you determine the common differences of the terms?