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Sequence

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Sequence

Study Guide

Key Definition

A sequence is a list of numbers in a specific order. Each number in a sequence is called a term. The n-th term, denoted as $a_n$, can often be expressed using a formula. For example, the sequence $\{5, 15, 25, 35, \ldots\}$ can be represented by $a_n = 10n - 5$.

Important Notes

A finite sequence has a limited number of terms.
An infinite sequence continues indefinitely.
An increasing sequence satisfies $a_{n+1} > a_n$.
A decreasing sequence satisfies $a_{n+1} < a_n$.
The position of a term is important for identifying the n-th term.

Mathematical Notation

$+$ represents addition

$-$ represents subtraction

$\times$ or $*$ represents multiplication

$\div$ or $/$ represents division

$a_n$ denotes the n-th term of the sequence

Remember to use proper notation when solving problems

Why It Works

Sequences represent ordered numbers where each term is determined by its position. Using formulas like $a_n = 10n - 5$, we can find any term in a sequence by plugging in the position number.

Remember

To find the n-th term of a sequence, identify the pattern or formula that describes the sequence.

Quick Reference

Finite Sequence Example:$\{4, 8, 12, \ldots, 64\}$

Infinite Sequence Example:$\{4, 8, 12, 16, \ldots\}$

Understanding Sequence

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Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

A sequence is a list of numbers in a specific order, such as $\{5, 15, 25, 35, \ldots\}$.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is the fifth term of the sequence $\{5, 15, 25, 35, \ldots\}$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

A sequence represents the savings each month: $\{20, 40, 60, \ldots\}$. How much is saved in the sixth month?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

In the sequence $\{5, 15, 25, \ldots\}$, which term is 85?

Challenge Quiz

Single Choice Quiz

Advanced

Which formula represents the n-th term of the sequence $\{7, 14, 21, \ldots\}$?

Recap

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