The difference between any two consecutive terms in an arithmetic sequence is called the common difference.
Understanding Arithmetic Sequences and the Common Difference
An arithmetic sequence is defined by a constant difference between successive terms. This constant difference is crucial for understanding the sequence and is known as the common difference.
- Definition: An arithmetic sequence is a sequence of numbers such that the difference between any two consecutive terms is constant.
- Common Difference: According to the reference, the common difference is the number added to any one term of an arithmetic sequence that generates the subsequent term.
Examples of Common Difference
Here are a few examples to illustrate the common difference:
- Sequence: 2, 4, 6, 8, 10...
- Common Difference: 2 (Each term is 2 more than the previous term.)
- Sequence: 1, 5, 9, 13, 17...
- Common Difference: 4 (Each term is 4 more than the previous term.)
- Sequence: 10, 7, 4, 1, -2...
- Common Difference: -3 (Each term is -3 more than the previous term.)
How to Find the Common Difference
To find the common difference, simply subtract any term from the term that follows it.
- Formula: d = an+1 - an, where 'd' is the common difference, an+1 is the (n+1)th term, and an is the nth term.
Importance of the Common Difference
The common difference is fundamental in working with arithmetic sequences:
- It allows you to predict future terms in the sequence.
- It is used in formulas for calculating the sum of arithmetic series.