The nth term of an arithmetic progression with first term a and common difference d is given by the formula a + (n - 1)d. This formula directly calculates any term in the sequence without needing to know the preceding terms.
Understanding Arithmetic Progressions
An arithmetic progression (AP) is a sequence of numbers such that the difference between any two consecutive terms is constant. This constant difference is called the common difference, denoted by d. The first term of the sequence is usually denoted by a.
Formula for the nth Term
The formula to find the nth term of an arithmetic progression is:
- an = a + (n - 1)d
Where:
- an is the nth term.
- a is the first term.
- n is the term number (e.g., 1 for the first term, 2 for the second term, and so on).
- d is the common difference.
Example
Let's consider an arithmetic progression where the first term (a) is 2 and the common difference (d) is 3. We want to find the 5th term (a5).
Using the formula:
a5 = a + (5 - 1)d
a5 = 2 + (4)3
a5 = 2 + 12
a5 = 14
Therefore, the 5th term of this arithmetic progression is 14.
Practical Insights
- Finding a Specific Term: This formula is particularly useful when you need to find a specific term in a large arithmetic progression without having to list all the preceding terms.
- Determining the Common Difference: If you know two terms in the sequence (say, the mth and nth terms), you can rearrange the formula to find the common difference, d.
- Applications: Arithmetic progressions have applications in various fields, including finance (simple interest), physics (motion with constant acceleration), and computer science (looping and indexing).
Table Summarizing Key Variables
| Variable | Description |
|---|---|
| a | First term of the AP |
| d | Common difference of the AP |
| n | Term number you want to find |
| an | The nth term of the AP |
