How do you find the nth value of a sequence?

To find the nth value of a sequence, you'll need to understand the type of sequence you're dealing with and then apply the relevant formula or method. There are different formulas and methods for different types of sequences. Here, we will focus on finding the nth term for an arithmetic sequence.

Finding the nth Term of an Arithmetic Sequence

An arithmetic sequence is a sequence of numbers where the difference between any two consecutive terms is constant. This constant difference is called the common difference.

The Formula

The formula to find the nth term (denoted as a_n) of an arithmetic sequence is:

a_n = a₁ + (n - 1)d

Where:

• a_n is the nth term that you want to find.
• a₁ is the first term of the sequence.
• n is the term number (the position of the term in the sequence).
• d is the common difference between consecutive terms.

Steps to Use the Formula

1. Identify the first term (a₁): Look at your sequence and identify the first number in the sequence.
2. Determine the common difference (d): Subtract any term from the next consecutive term. This should be a constant value for an arithmetic sequence.
3. Identify the term number (n): You need to know which term in the sequence you want to find (e.g., the 5th term, the 10th term, etc.).
4. Plug the values into the formula: Substitute the values of a₁, n, and d into the formula: a_n = a₁ + (n - 1)d
5. Solve for a_n: Perform the calculations to find the value of the nth term.

Example

Let's consider the arithmetic sequence: 2, 5, 8, 11, 14... and let's find the 7th term.

1. a₁ (first term) = 2
2. d (common difference) = 5 - 2 = 3 (or 8 - 5 = 3, etc.)
3. n (term number) = 7 (We want to find the 7th term)

Now, we use the formula:
a₇ = 2 + (7 - 1) 3
a₇ = 2 + (6) 3
a₇ = 2 + 18
a₇ = 20

Therefore, the 7th term of the sequence is 20.

Summary

The formula a_n = a₁ + (n - 1)d is a powerful tool for finding any term in an arithmetic sequence. By understanding what each element represents and following the steps, you can easily calculate the value of any nth term. This method is fundamental for working with arithmetic sequences. Remember to always verify that the sequence is actually arithmetic before applying this formula.