Geometric sequence
A geometric sequence is a Grade 7 number pattern in Saxon Math, Course 2 where each term is found by multiplying the previous term by the same constant factor (called the common ratio). In 1, 2, 4, 8, 16, ..., each term doubles, so the common ratio is 2. In 100, 50, 25, ..., the ratio is 1/2 (halving each time). Unlike arithmetic sequences where terms are equally spaced, geometric sequences grow exponentially — making them the foundation for understanding compound interest, population growth, and exponential functions.
Key Concepts
Property In a geometric sequence , each term is multiplied by the same number to find the next term. Terms in a geometric sequence are not equally spaced on a number line.
Examples The sequence $1, 2, 4, 8, 16, ...$ is geometric because you multiply by 2 each time to get the next term. In the sequence $100, 50, 25, ...$, the rule is to multiply by $\frac{1}{2}$, so the next term is $12.5$. The sequence $1, 10, 100, ...$ is geometric. The rule is 'multiply by 10', so the next term is $100 \cdot 10 = 1000$.
Explanation If an arithmetic sequence is like walking, a geometric sequence is like a rocket launch! Instead of adding, you multiply by the same number each time, causing the terms to grow (or shrink) incredibly fast. It’s all about multiplication, making for an explosive pattern where numbers can get huge or tiny in just a few steps.
Common Questions
What is a geometric sequence?
A geometric sequence is a list of numbers where each term is found by multiplying the previous term by the same constant value, called the common ratio.
How do you find the common ratio of a geometric sequence?
Divide any term by the term before it. If the result is consistent throughout the sequence, that value is the common ratio.
What is the difference between an arithmetic and a geometric sequence?
Arithmetic sequences add/subtract the same value each time (equally spaced on a number line). Geometric sequences multiply by the same value each time (not equally spaced).
Can a geometric sequence decrease?
Yes. If the common ratio is a fraction between 0 and 1 (like 1/2), the sequence decreases. For example, 100, 50, 25, 12.5, ... has ratio 1/2.
Where are geometric sequences taught in Saxon Math Course 2?
Geometric sequences are introduced in Saxon Math, Course 2, as part of Grade 7 patterns and number theory content.
What real-world situations follow geometric sequences?
Compound interest, bacterial population doubling, radioactive decay, and viral social media sharing all follow geometric sequence patterns.
How do you find the next term in a geometric sequence?
Multiply the last known term by the common ratio. For example, in 3, 6, 12, 24, ..., the ratio is 2, so the next term is 24 × 2 = 48.