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Sum of the First n Terms of a Geometric Sequence

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Sum of the First n Terms of a Geometric Sequence

Study Guide

Key Definition

The sum of the first $n$ terms of a geometric sequence 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 when $r \neq 1$.
For an infinite geometric series, the sum is $S = \frac{a_1}{1-r}$ if $|r| < 1$.
The common ratio $r$ is found by dividing any term by the previous term.
If $|r| > 1$, the series does not converge to a sum.
Infinite series sums are only possible when $|r| < 1$.

Mathematical Notation

$+$ represents addition

$-$ represents subtraction

$\times$ or $*$ represents multiplication

$\div$ or $/$ represents division

$\frac{a}{b}$ represents a fraction

Remember to use proper notation when solving problems

Why It Works

The formula $S_n = a_1 \frac{1-r^n}{1-r}$ works because it simplifies the process of adding up a geometric series by leveraging the powers of the common ratio.

Remember

Always check if $|r| < 1$ when calculating the sum of an infinite series.

Quick Reference

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

Infinite Series Formula:$S = \frac{a_1}{1-r}$ (if $|r| < 1$)

Understanding Sum of the First n Terms of a Geometric Sequence

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Video explanation of this concept

Beginner

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Beginner Explanation

Use $S_n = a_1 \frac{1-r^n}{1-r}$ to quickly add the first $n$ terms of a geometric sequence.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is the sum of the first 8 terms of a geometric sequence where $a_1 = 1$ and $r = 2$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

A phone subscription plan provides 24 MB of data that halves every month. How much total data will you have used after 10 months?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Consider an infinite series where the first term is 3 and the common ratio is $-\frac{1}{3}$. What is the sum?

Challenge Quiz

Single Choice Quiz

Advanced

For the sequence $24, 12, 6, \ldots$, find $S_{10}$.

Recap

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