Arithmetic Sequences and Patterns

Have you ever noticed a pattern in a list of numbers, like counting by twos or fives? When a sequence of numbers increases or decreases by the exact same amount every time, it is called an arithmetic sequence.

What is the Common Difference?

The fixed value that we add to get from one term to the next is called the common difference, represented by the letter $d$.

To find the common difference, simply subtract any term from the term that comes right after it: $d = a_2 - a_1$

The $n$th Term Formula

If you want to find a number way down the line (like the 50th term), adding the common difference over and over would take too long! Instead, we use the $n$th term formula:

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

Here is what each part means:

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

Example Problems

Example 1: Find the common difference and 10th term of the sequence 3, 7, 11, 15, ...

1. Find $d$: Subtract the first term from the second term. $d = 7 - 3 = 4$
2. Identify $a_1$: The first term is $3$.
3. Use the formula to find $a_{10}$: $a_{10} = 3 + (10 - 1)(4)$ $a_{10} = 3 + (9)(4)$ $a_{10} = 3 + 36 = 39$

The common difference is $4$, and the 10th term is $39$.

Example 2: Write the first 5 terms of an arithmetic sequence with first term 2 and common difference 5.

• Term 1 ($a_1$): $2$
• Term 2 ($a_2$): $2 + 5 = 7$
• Term 3 ($a_3$): $7 + 5 = 12$
• Term 4 ($a_4$): $12 + 5 = 17$
• Term 5 ($a_5$): $17 + 5 = 22$

The first 5 terms are 2, 7, 12, 17, 22.