You can use the formula for the nth term of an arithmetic sequence to find any specific term. The formula is: an = a1 + (n − 1)d. This formula directly allows you to calculate the value of any term within the sequence.
Understanding the Formula
- an: Represents the nth term, which is the specific term you're trying to find.
- a1: Represents the first term in the arithmetic sequence.
- n: Represents the term number; for example, if you want the 5th term, n would be 5.
- d: Represents the common difference between consecutive terms in the sequence.
Using the Formula to Find a Specific Term
Here's how to use the formula:
- Identify the First Term (a1): Determine the first number in your arithmetic sequence.
- Determine the Common Difference (d): Calculate the difference between any two consecutive terms (e.g., the second term minus the first term).
- Identify the Term Number (n): Determine which term you are looking for (e.g., the 10th term means n=10).
- Substitute and Solve: Substitute the values of a1, n, and d into the formula an = a1 + (n − 1)d and solve for an.
Example
Let's say you have an arithmetic sequence: 2, 5, 8, 11... and you want to find the 10th term.
- a1 (first term) = 2
- d (common difference) = 5 - 2 = 3
- n (term number) = 10
Substituting into the formula:
- a10 = 2 + (10 - 1) * 3
- a10 = 2 + (9) * 3
- a10 = 2 + 27
- a10 = 29
Therefore, the 10th term in the sequence is 29.
Practical Insights
- Flexibility: The formula is highly versatile and can be used to find any term in the sequence, no matter how far down the line it is.
- Efficiency: It's more efficient than manually finding each term by repeatedly adding the common difference.
- Direct Calculation: It allows you to directly calculate the value of a term if you know the first term and common difference.
- Reference: As per the provided reference on 21-Sept-2023, the formula an=a1+(n−1)d is indeed the correct method for finding the nth term in an arithmetic sequence.
