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Sum of the First n Terms of a Geometric Series

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Sum of the First n Terms of a Geometric Series

Study Guide

Key Definition

The sum of the first $n$ terms of a geometric series is given by $S_n = a_1 \frac{1 - r^n}{1 - r}$, where $a_1$ is the first term and $r$ is the common ratio.

Important Notes

The formula $S_n = a_1 \frac{1 - r^n}{1 - r}$ is only valid for $r \neq 1$.
A geometric series is defined by a constant ratio between successive terms.
If |r| < 1, the series converges as $n$ approaches infinity.
The sum formula simplifies calculations without adding each term individually.
Understanding the common ratio is key to solving geometric series problems.

Mathematical Notation

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

$a_1$ is the first term of the series

$r$ is the common ratio

$n$ is the number of terms

$\frac{1}{1 - r}$ is used in the sum formula

Remember to use proper notation when solving problems

Why It Works

The formula $S_n = a_1 \frac{1 - r^n}{1 - r}$ leverages the properties of geometric progression to simplify the sum of terms using the common ratio.

Remember

Always check the value of $r$ to ensure the series is geometric and use the formula accordingly.

Quick Reference

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

Understanding Sum of the First n Terms of a Geometric Series

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Video explanation of this concept

Beginner

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Beginner Explanation

A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio, $r$. The sum of the first $n$ terms is calculated using $S_n = a_1 \frac{1 - r^n}{1 - r}$.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is the sum of the first 3 terms of a geometric series where $a_1 = 2$ and $r = 3$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

Imagine saving money where each month you save 3 times the amount of the previous month, starting with $5. Calculate the total savings after 4 months.

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

If the sum of a geometric series is 75 and $a_1 = 5$ with $r = 2$, find the number of terms, $n$.

Challenge Quiz

Single Choice Quiz

Advanced

For a geometric series with $a_1 = 10$ and $r = 0.5$, what is the sum of the first 6 terms?

Recap

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