Arithmetic Sequences

An arithmetic sequence is an ordered list of numbers where the difference between any two consecutive terms is always the same. This constant value is called the common difference, represented by the letter $d$.

If you know the starting number and the common difference, you can build the entire sequence by repeatedly adding $d$.

The $n$th Term Formula

Instead of writing out every single number to find a term far down the line, you can use the $n$th term formula:

$a_n = a_1 + (n - 1)d$

• $a_n$ is the $n$th term (the value of the term you are trying to find).
• $a_1$ is the first term of the sequence.
• $n$ is the position of the term in the sequence.
• $d$ is the common difference.

Example 1: Finding a Specific Term

Problem: Find the 20th term of the sequence $3, 7, 11, 15, \dots$

Solution:

1. Identify the first term ($a_1$): $a_1 = 3$.
2. Find the common difference ($d$): Subtract the first term from the second term. $d = 7 - 3 = 4$.
3. Use the formula for $n = 20$:

$a_{20} = 3 + (20 - 1)4$ $a_{20} = 3 + (19)(4)$ $a_{20} = 3 + 76 = 79$

The 20th term of the sequence is $79$.

Example 2: Writing a Formula

Problem: Write a formula for the arithmetic sequence where $a_1 = 5$ and $d = -3$.

Solution:

1. Start with the general formula: $a_n = a_1 + (n - 1)d$.
2. Plug in the given values:

$a_n = 5 + (n - 1)(-3)$

3. Simplify the equation by distributing the $-3$:

$a_n = 5 - 3n + 3$ $a_n = -3n + 8$

The formula for any term in this sequence is $a_n = -3n + 8$.