The common formula for finding the nth term of an arithmetic sequence is: an = a + (n-1)d.
Understanding the Arithmetic Sequence Formula
An arithmetic sequence is a series of numbers where the difference between any two consecutive terms is constant. This constant difference is known as the common difference, denoted by 'd'. The formula helps us find any term in the sequence without listing out all the terms. Let's break down the components of the formula:
- an: Represents the nth term of the sequence, which is the term you want to find.
- a: Represents the first term of the sequence.
- n: Represents the position of the term you want to find in the sequence.
- d: Represents the common difference between consecutive terms.
How to Use the Formula
- Identify the first term (a): Look for the first number in the sequence.
- Determine the common difference (d): Subtract any term from the term that follows it.
- Decide which term to find (n): Determine which position's value you are calculating.
- Apply the formula: Substitute the values of a, n, and d into the formula: an = a + (n-1)d
- Calculate: Solve the equation to find the value of the nth term.
Example
Let’s consider an arithmetic sequence: 2, 5, 8, 11, ...
- a (the first term) is 2.
- d (the common difference) is 5 - 2 = 3.
- Let’s find the 5th term (n = 5).
Using the formula:
a5 = 2 + (5 - 1) * 3
a5 = 2 + (4) * 3
a5 = 2 + 12
a5 = 14
Therefore, the 5th term of this sequence is 14.
Practical Insights
- The formula is crucial for quickly calculating any term in the sequence, especially when dealing with large sequences or finding terms far down the list.
- Arithmetic sequences are common in various real-world applications, such as calculating compound interest, predicting equipment depreciation and scheduling recurring tasks.
| Component | Description |
|---|---|
| an | The nth term of the arithmetic sequence |
| a | The first term of the sequence |
| n | The position of the term to be determined |
| d | The common difference between consecutive terms |
