Geometric Sequence Study Guide

What is a 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. The general form of a geometric sequence is:
a, ar, ar^2, ar^3, ..., where a is the first term and r is the common ratio.

Finding the nth Term of a Geometric Sequence

The nth term of a geometric sequence can be found using the formula:
an = a * r^(n-1), where an is the nth term, a is the first term, r is the common ratio, and n is the term number.

Finding the Sum of the First n Terms of a Geometric Sequence

The sum of the first n terms of a geometric sequence can be found using the formula:
Sn = a * (1 - r^n) / (1 - r), where Sn is the sum of the first n terms, a is the first term, r is the common ratio, and n is the number of terms.

Common Ratio

The common ratio (r) is the ratio between consecutive terms in a geometric sequence. It is found by dividing any term by the previous term. The common ratio is an important factor in determining the behavior of a geometric sequence.

Example

For example, if the first term (a) of a geometric sequence is 3 and the common ratio (r) is 2, then the sequence would be:
3, 6, 12, 24, 48, ...

Conclusion

Understanding geometric sequences and how to find the nth term and the sum of the first n terms is important for various mathematical and real-world applications. Practice solving problems and working with geometric sequences to solidify your understanding of the topic.

Summary

• A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed non-zero number called the common ratio.
• The nth term of a geometric sequence can be found using the formula: an = a * r^(n-1).
• The sum of the first n terms of a geometric sequence can be found using the formula: Sn = a * (1 - r^n) / (1 - r).
• The common ratio (r) is the ratio between consecutive terms in a geometric sequence.