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Geometric Sequences

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Geometric Sequences

Study Guide

Key Definition

A geometric sequence is defined by its initial term and a common ratio. Each term after the first is the previous term multiplied by the common ratio $r$.

Important Notes

In a geometric sequence, the ratio between consecutive terms is constant.
If the common ratio $r$ is between 0 and 1, the terms decrease.
If $r > 1$, the terms increase.
$r = 1$ results in a constant sequence.
Geometric sequences can model compound interest.

Mathematical Notation

$a_n$ represents the nth term of the sequence

$r$ is the common ratio

$a_1$ is the initial term

$n$ is the term number

$a_n = a_1 × r^{n-1}$ is the formula for the nth term

Remember to use proper notation when solving problems

Why It Works

A geometric sequence allows us to model exponential growth or decay using $r$.

Remember

The formula for the nth term of a geometric sequence is $a_n = a_1 × r^{n-1}$.

Quick Reference

Formula for nth Term:$a_n = a_1 × r^{n-1}$

Understanding Geometric Sequences

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Video explanation of this concept

Beginner

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Beginner Explanation

A geometric sequence is a list where each term is found by multiplying the previous term by a fixed number called the common ratio. For example, in $2, 6, 18, ...$ each term is multiplied by $3$. The formula for the $n$th term is $a_n = a_1 × r^{n-1}$.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is the common ratio in the sequence $2, 6, 18, 54, ...$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

You invest $$1000$ in a bank with an annual interest rate of $5%$. Find the amount after 3 years, assuming interest compounds annually.

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Consider a bouncy ball dropped from $10$ feet. Each bounce reaches $0.6$ of the previous height. How high is the ball on the third bounce?

Challenge Quiz

Single Choice Quiz

Advanced

Find the fifth term of the sequence $5, 15, 45, ...$.

Recap

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