Geometric Sequence

Definition

A geometric sequence is a sequence where each term is obtained by multiplying the previous term by a constant ratio. This constant ratio is known as the common ratio, and it determines the pattern of the sequence.

5 Must Know Facts For Your Next Test

1. The general formula for a geometric sequence is $a_n = a_1 \cdot r^{n-1}$, where $a_1$ is the first term and $r$ is the common ratio.
2. If the common ratio $r$ is greater than 1, the sequence is said to be an increasing geometric sequence. If $r$ is less than 1, the sequence is a decreasing geometric sequence.
3. Geometric sequences have many applications in real-world scenarios, such as compound interest, population growth, and radioactive decay.
4. The sum of the first $n$ terms of a geometric sequence can be calculated using the formula $S_n = a_1 \cdot \frac{1 - r^n}{1 - r}$.
5. Geometric series, which are the sum of an infinite number of terms in a geometric sequence, have important applications in mathematics and physics.

Review Questions

• Explain the relationship between the common ratio and the behavior of a geometric sequence.
  • The common ratio, denoted as $r$, is the constant ratio between consecutive terms in a geometric sequence. If the common ratio $r$ is greater than 1, the sequence is an increasing geometric sequence, meaning each term is larger than the previous term. If the common ratio $r$ is less than 1, the sequence is a decreasing geometric sequence, meaning each term is smaller than the previous term. The value of the common ratio directly determines the pattern and behavior of the geometric sequence.
• Describe how to calculate the sum of the first $n$ terms of a geometric sequence.
  • The sum of the first $n$ terms of a geometric sequence can be calculated using the formula $S_n = a_1 \cdot \frac{1 - r^n}{1 - r}$, where $a_1$ is the first term and $r$ is the common ratio. This formula is derived from the fact that each term in a geometric sequence is a constant multiple of the previous term. By summing the terms using this formula, you can efficiently calculate the total sum of the first $n$ terms of the sequence.
• Analyze the relationship between geometric sequences and geometric series, and explain their practical applications.
  • Geometric sequences and geometric series are closely related, as a geometric series is the sum of an infinite number of terms in a geometric sequence. Geometric series have important applications in various fields, such as finance (e.g., compound interest), physics (e.g., radioactive decay), and engineering (e.g., signal processing). Understanding the properties and formulas of geometric sequences and series allows for the modeling and analysis of these real-world phenomena, which is crucial for making accurate predictions and informed decisions.

Related terms

Sequence: A sequence is an ordered list of numbers or objects, where each item in the list is called a term.

Common Ratio: The common ratio is the constant ratio between consecutive terms in a geometric sequence.

Arithmetic Sequence: An arithmetic sequence is a sequence where the difference between consecutive terms is constant, unlike a geometric sequence where the ratio between consecutive terms is constant.