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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 sequenceRemember 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

concept. Use space or enter to play video.concept thumbnailBeginner

Start here! Easy to understand

BeginnerIntermediateAdvanced

Beginner Explanation

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

Practice Problems

Test your understanding with practice problems

1

Quick Quiz

Single Choice QuizBeginner

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

A$45$B$50$C$55$D$60$Check AnswerPlease select an answer for all 1 questions before checking your answers. 1 question remaining.2

Real-World Problem

Question ExerciseIntermediate

Teenager Scenario

A sequence represents the savings each month: $\{20, 40, 60, \ldots\}$. How much is saved in the sixth month?Show AnswerClick to reveal the detailed solution for this question exercise.3

Thinking Challenge

Thinking ExerciseIntermediate

Think About This

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

Show AnswerClick to reveal the detailed explanation for this thinking exercise.4

Challenge Quiz

Single Choice QuizAdvanced

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

A$a_n = 7n$B$a_n = 7n + 7$C$a_n = 7n - 7$D$a_n = n + 7$Check AnswerPlease select an answer for all 1 questions before checking your answers. 1 question remaining.

Recap

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Review key concepts and takeaways

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