Finding the Common Difference

Oct 13, 2025 by talynt.org 121 views

Introduction

An arithmetic sequence is a sequence of numbers in which the difference between any two successive members is constant. In this article, we will consider the arithmetic sequence $11 \frac{1}{2}, 9 \frac{5}{16}, 7 \frac{1}{8}, \ldots$ and find the common difference between each term.

What is an Arithmetic Sequence?

An arithmetic sequence is a sequence of numbers in which the difference between any two successive members is constant. This constant difference is called the common difference. For example, the sequence $2, 4, 6, 8, \ldots$ is an arithmetic sequence with a common difference of $2$ .

Finding the Common Difference

To find the common difference of the given arithmetic sequence, we need to subtract each term from the previous term. Let's start by subtracting the second term from the first term:

\left(9 \frac{5}{16}\right) - \left(11 \frac{1}{2}\right) = -2 \frac{3}{16}

This means that the common difference between the first two terms is $-2 \frac{3}{16}$ .

Verifying the Common Difference

To verify that the common difference is indeed $-2 \frac{3}{16}$ , we can subtract the third term from the second term:

\left(7 \frac{1}{8}\right) - \left(9 \frac{5}{16}\right) = -2 \frac{3}{16}

As we can see, the common difference between the second and third terms is also $-2 \frac{3}{16}$ . This confirms that the common difference of the given arithmetic sequence is indeed $-2 \frac{3}{16}$ .

Conclusion

In conclusion, the common difference between each term of the arithmetic sequence $11 \frac{1}{2}, 9 \frac{5}{16}, 7 \frac{1}{8}, \ldots$ is $-2 \frac{3}{16}$ .

Answer

The correct answer is A. $-2 \frac{3}{16}$ .

Additional Examples

Here are a few more examples of arithmetic sequences and their common differences:

The sequence $3, 6, 9, 12, \ldots$ has a common difference of $3$ .
The sequence $5, 10, 15, 20, \ldots$ has a common difference of $5$ .
The sequence $2, 5, 8, 11, \ldots$ has a common difference of $3$ .

Tips and Tricks

Here are a few tips and tricks to help you find the common difference of an arithmetic sequence:

Start by subtracting the second term from the first term.
Verify the common difference by subtracting the third term from the second term.
Make sure to simplify your answer to its simplest form.

Common Mistakes

Here are a few common mistakes to avoid when finding the common difference of an arithmetic sequence:

Not simplifying your answer to its simplest form.
Not verifying the common difference by subtracting the third term from the second term.
Not using a calculator to simplify fractions.

Conclusion

Q: What is an arithmetic sequence?

A: An arithmetic sequence is a sequence of numbers in which the difference between any two successive members is constant. This constant difference is called the common difference.

Q: How do I find the common difference of an arithmetic sequence?

A: To find the common difference of an arithmetic sequence, you need to subtract each term from the previous term. For example, if the sequence is $2, 4, 6, 8, \ldots$ , you can find the common difference by subtracting each term from the previous term:

4 - 2 = 2

6 - 4 = 2

8 - 6 = 2

As you can see, the common difference is $2$ .

Q: What if the sequence has a negative common difference?

A: If the sequence has a negative common difference, it means that the terms are decreasing. For example, the sequence $10, 8, 6, 4, \ldots$ has a common difference of $-2$ .

Q: Can I use a calculator to find the common difference?

A: Yes, you can use a calculator to find the common difference. Simply subtract each term from the previous term and simplify the result to its simplest form.

Q: What if the sequence has a fraction as a common difference?

A: If the sequence has a fraction as a common difference, you can simplify the fraction to its simplest form. For example, the sequence $3 \frac{1}{2}, 2 \frac{1}{4}, 1 \frac{1}{8}, \ldots$ has a common difference of $-\frac{1}{2}$ .

Q: Can I find the common difference of a sequence with a variable?

A: Yes, you can find the common difference of a sequence with a variable. For example, the sequence $x, x + 2, x + 4, \ldots$ has a common difference of $2$ .

Q: What if the sequence has a common difference of zero?

A: If the sequence has a common difference of zero, it means that the terms are constant. For example, the sequence $5, 5, 5, 5, \ldots$ has a common difference of zero.

Q: Can I find the common difference of a sequence with a negative variable?

A: Yes, you can find the common difference of a sequence with a negative variable. For example, the sequence $-x, -x - 2, -x - 4, \ldots$ has a common difference of $-2$ .

Q: What if I'm given a sequence and asked to find the nth term?

A: If you're given a sequence and asked to find the nth term, you can use the formula for the nth term of an arithmetic sequence:

a_n = a_1 + (n - 1)d

where $a_n$ is the nth term, $a_1$ is the first term, $n$ is the term number, and $d$ is the common difference.

Q: Can I use the formula for the nth term to find the common difference?

A: Yes, you can use the formula for the nth term to find the common difference. Simply rearrange the formula to solve for $d$ :

d = \frac{a_n - a_1}{n - 1}

Conclusion

In conclusion, finding the common difference of an arithmetic sequence is a simple process that involves subtracting each term from the previous term. By following the tips and tricks outlined in this article, you can easily find the common difference of any arithmetic sequence.