Geometric Sequence

A geometric sequence is a sequence of numbers in which each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.

General Form

The general form of a geometric sequence is:

a, ar, ar², ar³, ar⁴, ...

Where:

• a = the first term
• r = the common ratio
• n = the position of the term in the sequence

Formula

The nth term of a geometric sequence can be found using the formula:

a_n = a * r^(n-1)

Sum of a Geometric Sequence

The sum of the first n terms of a geometric sequence can be found using the formula:

S_n = a * (1 - rⁿ) / (1 - r)

Study Guide

Here are some key points to remember when studying geometric sequences:

1. Understand the concept of a common ratio and how it is used to generate the terms of the sequence.
2. Be able to identify the first term (a) and the common ratio (r) in a given geometric sequence.
3. Practice finding specific terms of a geometric sequence using the formula a_n = a * r^(n-1).
4. Practice finding the sum of a certain number of terms in a geometric sequence using the formula S_n = a * (1 - rⁿ) / (1 - r).
5. Understand how to recognize and work with geometric sequences in real-world problems.

Remember to practice solving problems involving geometric sequences to reinforce your understanding of the topic.

Good luck with your studies!