Master

Recursive Sequence

Master recursive sequence with interactive lessons and practice problems! Designed for students like you!

Learn

Get the basics

Practice

Apply your skills

Master

Achieve excellence

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 multiplication

Remember 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

Choose your learning level

Watch & Learn

Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

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

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

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

Real-World Problem

Question Exercise

Intermediate

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?

Thinking Challenge

Thinking Exercise

Intermediate

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 $a_6$?

Challenge Quiz

Single Choice Quiz

Advanced

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

Recap

Watch & Learn

Review key concepts and takeaways