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Geometric Series

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Geometric Series

Study Guide

Key Definition

A geometric series is the sum of the terms of a geometric sequence. For a finite geometric series (assuming r \u2260 1), the sum is calculated using $S_n = a_1 \frac{1 - r^n}{1 - r}$ where $a_1$ is the first term, $r$ is the common ratio, and $n$ is the number of terms.

Important Notes

The common ratio $r$ determines the progression of the sequence.
A finite geometric series has a limited number of terms.
An infinite geometric series continues indefinitely.
If $|r| < 1$, the infinite series converges.
Use sigma notation to represent series compactly.
If $|r| \u2265 1$, the infinite series diverges.

Mathematical Notation

$\Sigma$ represents summation

$r^n$ represents the nth power of the common ratio

$a_1$ is the first term of the sequence

$r$ is the common ratio

$S_n$ is the sum of the first $n$ terms

Remember to use proper notation when solving problems

Why It Works

Start with S_n = a_1 + a_1r + a_1r^2 + \u2026 + a_1r^{n-1}. Multiply both sides by r to get r S_n, then subtract: S_n - r S_n = a_1 - a_1r^n. Solve for S_n to obtain the sum formula.

Remember

The key formula for a finite geometric series: $S_n = a_1 \frac{1 - r^n}{1 - r}$

Quick Reference

Sum Formula:$S_n = a_1 \frac{1 - r^n}{1 - r}$

Understanding Geometric Series

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Video explanation of this concept

Beginner

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Beginner Explanation

A geometric series is the sum of terms in a geometric sequence, where each term is multiplied by a constant ratio. For example, the series $1 + \frac{1}{2} + \frac{1}{4}$.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is the sum of the first three terms of the geometric sequence $1, 2, 4$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

Imagine saving money each week, doubling the amount you save from the previous week, starting with $1. How much money will you have saved in total after 4 weeks?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

If a geometric series starts with $3$ and has a common ratio of $\frac{1}{2}$, calculate the sum of the first five terms.

Challenge Quiz

Single Choice Quiz

Advanced

Find the sum of the infinite geometric series $2 + 1 + \frac{1}{2} + \frac{1}{4} + \ldots$

Recap

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Review key concepts and takeaways