What is the general rule formula for arithmetic sequence?

The general rule formula for an arithmetic sequence allows you to find any term in the sequence.

Understanding Arithmetic Sequences

An arithmetic sequence is a sequence where the difference between consecutive terms is constant. This constant difference is known as the common difference. Knowing the first term and the common difference lets you calculate any term in the sequence.

The Formula

The formula for the nth term (a_n) of an arithmetic sequence is:

a_n = a₁ + (n - 1)d

Where:

• a_n is the nth term you want to find.
• a₁ is the first term of the sequence.
• n is the position of the term in the sequence (e.g., 3rd term, 10th term).
• d is the common difference between consecutive terms.

According to reference information, the formula for the nth term in an arithmetic sequence is a_n = a₁ + (n - 1)d, which can be used to determine the value of any term in an arithmetic sequence.

Example

Let's say we have an arithmetic sequence: 2, 5, 8, 11, ...

• a₁ (the first term) = 2
• d (the common difference) = 3 (because 5-2 = 3, 8-5 = 3, etc.)

To find the 10th term (a₁₀):

• n = 10

Using the formula:

• a₁₀ = 2 + (10 - 1) * 3
• a₁₀ = 2 + (9) * 3
• a₁₀ = 2 + 27
• a₁₀ = 29

Therefore, the 10th term of this arithmetic sequence is 29.

Importance of the Formula

This formula is incredibly useful because:

• It allows you to calculate any term directly without having to list out all the preceding terms.
• It simplifies the process of analyzing and understanding arithmetic sequences.