Recursive Sequence - Varsity Tutors

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Recursive Sequence

Study Guide

Key Definition

A recursive sequence is a sequence where each term is defined as a function of one or more of the preceding terms. For an arithmetic sequence, $a_{n+1} = a_n + d$, and for a geometric sequence, $a_{n+1} = a_n \times r$.

Important Notes

  • You need to know the previous term(s) to find the next term.
  • For arithmetic sequences, use $a_{n+1} = a_n + d$.
  • For geometric sequences, use $a_{n+1} = a_n \times r$.
  • The common difference $d$ is key for arithmetic sequences.
  • The common ratio $r$ is essential for geometric sequences.

Mathematical Notation

$a_n$ represents the nth term$d$ is the common difference in arithmetic sequences$r$ is the common ratio in geometric sequences$+$ represents addition$\times$ represents multiplicationRemember to use proper notation when solving problems

Why It Works

Recursive formulas build each term based on the previous ones, allowing for step-by-step construction of sequences.

Remember

Always start with known terms and apply the recursive formula to find subsequent terms.

Quick Reference

Arithmetic Sequence Formula:$a_{n+1} = a_n + d$Geometric Sequence Formula:$a_{n+1} = a_n \times r$

Understanding Recursive 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

An arithmetic sequence has a constant difference, $d$, between terms. Use $a_{n+1} = a_n + d$.Now showing Beginner level explanation.

Practice Problems

Test your understanding with practice problems

1

Quick Quiz

Single Choice QuizBeginner

Given the arithmetic sequence where $a_1 = 5$ and $d = 3$, what is $a_4$?

A$11$B$12$C$13$D$14$Check AnswerPlease select an answer for all 1 questions before checking your answers. 1 question remaining.2

Real-World Problem

Question ExerciseIntermediate

Teenager Scenario

If you start with $2$ pieces of candy and double the amount each day, how many pieces will you have after 4 days?Show AnswerClick to reveal the detailed solution for this question exercise.3

Thinking Challenge

Thinking ExerciseIntermediate

Think About This

Consider a sequence where each term is the sum of the previous two terms, starting with $a_1 = 1$ and $a_2 = 1$. What is $a_6$?

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

Challenge Quiz

Single Choice QuizAdvanced

In a geometric sequence, $a_3 = 18$ and $r = 3$. What is $a_5$?

A$54$B$162$C$243$D$486$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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