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Arithmetic Sequences

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Arithmetic Sequences

Study Guide

Key Definition

An arithmetic sequence is a pattern of numbers in which the difference between any two consecutive terms stays the same. This difference is often referred to as the <a href="common-difference">$d$</a>.

Important Notes

In an arithmetic sequence, the difference between any two consecutive terms is constant.
This constant difference can be positive or negative.
If you need to find a specific term in the sequence, you can use the formula $a_n = a_1 + (n-1)d$ where $a_n$ is the nth term, $a_1$ is the first term of the sequence, and <a href="common-difference">$d$</a> is the common difference.

Mathematical Notation

$a_n$ denotes the nth term of the sequence.

$a_1$ denotes the first term of the sequence.

<a href="common-difference">$d$</a> denotes the common difference of the sequence.

Remember to use proper notation when solving problems about arithmetic sequences.

Why It Works

The arithmetic sequence formula works because it is based on the property that the difference between consecutive terms is constant. This constant difference makes it so that each term can be calculated by adding the common difference to the previous term.

Remember

The formula for the nth term of an arithmetic sequence is $a_n = a_1 + (n-1)d$.

Quick Reference

Arithmetic Sequence Formula:$a_n = a_1 + (n-1)d$

Understanding Arithmetic Sequences

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Video explanation of this concept

Beginner

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Beginner Explanation

An arithmetic sequence is a pattern of numbers in which the difference between any two consecutive terms stays the same. This difference is known as the common difference.

Practice Problems

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Quick Quiz

Single Choice Quiz

Beginner

What is the common difference in the following sequence: $5, 10, 15, 20, 25...$?

Real-World Problem

Question Exercise

Intermediate

Baking Scenario

If a recipe calls for adding $2$ teaspoons of sugar every $5$ minutes to a sauce, starting immediately at time 0 and then every 5 minutes thereafter, how many teaspoons of sugar will have been added after $20$ minutes?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Find the 10th term of the arithmetic sequence $3, 6, 9, 12, 15...$

Challenge Quiz

Single Choice Quiz

Advanced

What is the 20th term in the following sequence: $10, 7, 4, 1, -2...$?

Recap

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