How Do You Read an Arithmetic Sequence?

An arithmetic sequence is read by identifying the first term and then noting the constant difference between each subsequent term.

Here's a breakdown of how to read and understand an arithmetic sequence:

• Identify the Sequence: An arithmetic sequence is a list of numbers where the difference between any two consecutive terms is constant. This constant difference is called the "common difference."

• Find the First Term (a₁): This is simply the first number in the sequence.

• Determine the Common Difference (d): Subtract any term from the term that follows it. For example, subtract the first term from the second term (a₂ - a₁), or the second term from the third term (a₃ - a₂), and so on. If the result is the same each time, you have an arithmetic sequence.

• Express the Sequence: You can then express the arithmetic sequence as: a₁, a₁ + d, a₁ + 2d, a₁ + 3d, ...

Example:

Consider the sequence: 2, 5, 8, 11, 14, ...

1. First Term (a₁): 2
2. Common Difference (d): 5 - 2 = 3. Also, 8 - 5 = 3, 11 - 8 = 3, and so on. The common difference is 3.
3. Reading the Sequence: "The arithmetic sequence starts at 2, and each subsequent term is 3 more than the previous term."

General Formula for the nth Term (a_n):

The nth term (a_n) of an arithmetic sequence can be found using the formula:

a_n = a₁ + (n - 1)d

Where:

• a_n is the nth term
• a₁ is the first term
• n is the term number (e.g., 1st, 2nd, 3rd, ...)
• d is the common difference

Example Using the Formula:

Using the sequence 2, 5, 8, 11, 14, ... Let's find the 5th term (a₅) using the formula:

• a₁ = 2
• d = 3
• n = 5

a₅ = 2 + (5 - 1) 3
a₅ = 2 + (4) 3
a₅ = 2 + 12
a₅ = 14

This confirms that the 5th term in the sequence is indeed 14.

By identifying the first term and the common difference, you can easily understand and predict any term in an arithmetic sequence.