How to Find a n in an Arithmetic Sequence?

To find a_n, which represents a specific term in an arithmetic sequence, you can use the arithmetic sequence formula. Here’s how:

Understanding the Arithmetic Sequence Formula

The arithmetic sequence formula is given by:

• a_n = a₁ + (n - 1)d

Where:

• a_n is the nth term of the sequence (what we want to find).
• a₁ is the first term of the sequence.
• n is the number of the term you're looking for (e.g., if you want the 5th term, n=5).
• d is the common difference between consecutive terms.

Identifying n

As stated in the reference, the variable 'n' in the arithmetic sequence formula is the number of the term you're trying to find. For example, if you are trying to determine the 2nd term of a sequence, n would be equal to 2.

Steps to Find a_n

1. Identify a₁: Determine the first term of the arithmetic sequence.
2. Identify d: Calculate the common difference (d) by subtracting any term from the term that follows it.
   • Example: If the sequence is 2, 5, 8, 11..., the common difference (d) is 5 - 2 = 3 or 8-5 =3 etc.
3. Identify n: Determine which term you want to find.
   • Example: For the 10th term, n is 10.
4. Substitute: Plug the values of a₁, n, and d into the arithmetic sequence formula: a_n = a₁ + (n - 1)d
5. Calculate: Solve the equation to find the value of a_n.

Example

Let's find the 10th term (a₁₀) of the sequence 2, 5, 8, 11...

1. a₁ = 2
2. d = 3
3. n = 10
4. Substitute: a₁₀ = 2 + (10 - 1)3
5. Calculate: a₁₀ = 2 + (9)3 = 2 + 27 = 29

Therefore, the 10th term (a₁₀) of the sequence is 29.

Table Summary

Variable	Description
an	The nth term you're trying to find
a1	The first term of the sequence
n	The term number
d	The common difference between terms

In conclusion, finding a_n involves knowing the first term (a₁), the common difference (d), and the term number (n), and then using the arithmetic sequence formula a_n = a₁ + (n - 1)d.