Lesson 34: Recognizing and Extending Arithmetic Sequences

New Concept

An arithmetic sequence is a sequence that has a constant difference between two consecutive terms called the common difference.

What’s next

Next, you will learn the key formulas for describing these sequences. We'll use them to find any term and solve problems involving constant growth or change.

Property

An arithmetic sequence is a sequence that has a constant difference between two consecutive terms, which is called the common difference.

Examples

• The sequence $7, 12, 17, 22, \dots$ is arithmetic because the common difference is $5$.
• The sequence $7, 4, 1, -2, \dots$ is arithmetic because the common difference is $-3$.
• The sequence $3, 6, 12, 24, \dots$ is not arithmetic because the difference changes from $3$ to $6$.

Explanation

Think of it as marching with a steady beat! Each number is a step, and the 'common difference' is the fixed distance between each step. To see if a sequence is playing this tune, just subtract any term from the one that comes right after it and check if the result is always the same.

Property

Use the recursive formula $a_n = a_{n-1} + d$ to find the next term in a sequence, where $a_1$ is the first term, $d$ is the common difference, and $n$ is the term number.

Examples

• For a sequence with $a_1 = -2$ and $d=7$, the second term is $a_2 = -2 + 7 = 5$.
• Continuing that same sequence, the third term is found using the second term: $a_3 = 5 + 7 = 12$.
• And the fourth term uses the third: $a_4 = 12 + 7 = 19$. The sequence begins $-2, 5, 12, 19, \dots$

Explanation

This is the 'one step at a time' formula! It tells you how to get the next term ($a_n$) if you already know the term right before it ($a_{n-1}$). Just take your current term and add the common difference, $d$. It’s like taking one more predictable step on a staircase, over and over again.