The common difference in an arithmetic progression (AP) is found by subtracting any term from its immediate succeeding term.
Understanding Arithmetic Progression
An arithmetic progression is a sequence of numbers where the difference between any two consecutive terms is constant. This constant difference is known as the common difference.
How to Calculate Common Difference
To calculate the common difference, follow these steps:
- Identify Consecutive Terms: Select any two consecutive terms in the arithmetic progression.
- Subtract: Subtract the first term from the second term.
- Verify (Optional): To confirm, repeat this subtraction process with other consecutive pairs. If the result is the same, it is confirmed as the common difference of AP.
Formula
The common difference (d) can be expressed by the formula:
d = a₂ - a₁
where:
- a₂ is the second term in the AP
- a₁ is the first term in the AP
This concept can be generalized to:
d = aₙ - aₙ₋₁
where:
- aₙ is the nth term in the AP
- aₙ₋₁ is the (n-1)th term in the AP
Examples
-
Example 1: Consider the A.P: 2, 4, 6, 8...
- Here, a₁ = 2 and a₂ = 4.
- The common difference, d = 4 - 2 = 2.
- We can verify it with other consecutive terms as well: 6-4=2 and 8-6=2
-
Example 2: Consider the A.P: 10, 7, 4, 1...
- Here, a₁ = 10 and a₂ = 7
- The common difference, d = 7 - 10 = -3
Practical Insights
- The common difference can be positive, negative, or zero.
- Once you find the common difference, you can predict the subsequent terms of the arithmetic progression by adding the common difference to the last term in the sequence.
- Understanding the concept of common difference is essential for solving more complex arithmetic progression problems.
| Term | Value |
|---|---|
| First Term (a₁) | Value of the first term |
| Second Term (a₂) | Value of the second term |
| Common Difference (d) | a₂ - a₁ |
By subtracting any term from its subsequent term in the AP, you can find the common difference. This is a fundamental concept in arithmetic progression.
