How do you write a formula for the general term of an arithmetic sequence?

The general term of an arithmetic sequence can be written using a formula that expresses any term in the sequence based on its position.

Understanding Arithmetic Sequences

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

Formula for the General Term

The general term, often denoted as a_n, of an arithmetic sequence can be expressed as:

a_n = a₁ + (n - 1)d

Where:

• a_n is the nth term of the sequence.
• a₁ is the first term of the sequence.
• n is the position of the term in the sequence (1st, 2nd, 3rd, etc.).
• d is the common difference between consecutive terms.

Steps to find the formula

Here are the steps to find the formula for the general term:

1. Identify the first term (a₁): This is the first number in your arithmetic sequence.
2. Determine the common difference (d): Subtract any term from the term that follows it to find the consistent difference between the terms.
3. Plug a₁ and d into the general formula: Replace a₁ and d in the formula a_n = a₁ + (n - 1)d with the values you found.
4. Simplify the formula: Distribute and combine like terms to get a simplified equation.

Example

Let's say an arithmetic sequence starts with 7, and the common difference is 3.
Referencing the provided video, the sequence was developed as follows:

• a₁ = 7
• d = 3
• a_n = 7 + (n - 1) * 3

Simplifying, this becomes:

• a_n = 7 + 3n - 3
• a_n = 3n + 4

Therefore, the general term for this arithmetic sequence is a_n = 3n + 4.

Using the Formula

Once you have the formula, you can find any term in the sequence by substituting the desired term position for 'n'. For instance, to find the 10th term, you would use n=10.

• a₁₀ = 310 + 4
• a₁₀ = 30 + 4
• a₁₀ = 34

Thus, the 10th term of this sequence is 34.