To find the common difference of an arithmetic sequence, you subtract any term from its subsequent term.
Understanding the Common Difference
An arithmetic sequence is a list of numbers where each term after the first is obtained by adding a constant value. This constant value is called the common difference. It's the same difference between any two consecutive terms in the sequence.
Formula for Common Difference
The formula to find the common difference, d, is:
d = a(n) - a(n - 1)
Where:
- a(n) is any term in the sequence.
- a(n - 1) is the term immediately preceding a(n).
How to Calculate the Common Difference
- Identify any two consecutive terms: Choose any two terms in the arithmetic sequence where one immediately follows the other.
- Subtract the earlier term from the later term: Apply the formula: d = a(n) - a(n-1).
- The result is the common difference: This value should be the same between any two consecutive terms in the sequence.
Example
Let's look at the arithmetic sequence: 2, 5, 8, 11, 14...
- Using the first two terms: a(2) = 5, a(1) = 2, so d = 5 - 2 = 3
- Using the third and fourth terms: a(4) = 11, a(3) = 8, so d = 11 - 8 = 3
- Using the fourth and fifth terms: a(5) = 14, a(4) = 11, so d = 14 - 11 = 3
In each case, the common difference is 3.
Key Takeaways
- The common difference is constant throughout an arithmetic sequence.
- You can use any two consecutive terms to calculate it.
- The formula
d = a(n) - a(n - 1)is the general method for finding the common difference.
