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n-th Term of a Geometric Sequence

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n-th Term of a Geometric Sequence

Study Guide

Key Definition

The n-th term of a geometric sequence is given by $a_n = a_1 \cdot r^{n-1}$ where $a_1$ is the first term and $r$ is the common ratio.

Important Notes

The common ratio $r$ is found by dividing any term by the previous term.
A geometric sequence can be increasing or decreasing, depending on the value of $r$.
If $|r| < 1$, the sequence converges to zero.
If $r = 1$, the sequence is constant.
Understanding $a_n$ helps in predicting future values in the sequence.

Mathematical Notation

$\cdot$ represents multiplication

$+$ represents addition

$n$ is the term number

$a_1$ is the first term

$r$ is the common ratio

Remember to use proper notation when solving problems

Why It Works

The formula $a_n = a_1 \cdot r^{n-1}$ works because each term is multiplied by the common ratio repeatedly.

Remember

Always identify the first term and common ratio to use the formula $a_n = a_1 \cdot r^{n-1}$ effectively.

Quick Reference

Formula for n-th term:$a_n = a_1 \cdot r^{n-1}$

Understanding n-th Term of a Geometric Sequence

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Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

For example, in the sequence $2, 4, 8, 16, …$ we have $a_1 = 2$ and $r = 2$. To find the 5th term: $a_5 = 2 \cdot 2^{(5-1)} = 32$.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

Find the 4th term of the sequence $3, 9, 27, ...$

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

You save money and double your amount every month. If you start with $\$10$, how much will you have after 6 months (six doublings)?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Given a sequence where $a_2 = 16$ and $a_4 = 256$, find r.

Challenge Quiz

Single Choice Quiz

Advanced

Find the common ratio if $a_3 = 64$ and $a_6 = 512$

Recap

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Review key concepts and takeaways