Common Ratio

Definition

The common ratio is a constant ratio between consecutive terms in a geometric sequence. It represents the multiplicative relationship between each term in the sequence and the previous term.

5 Must Know Facts For Your Next Test

1. The common ratio is denoted by the variable $r$ and is calculated by dividing any term in the sequence by the previous term.
2. If the common ratio $r$ is greater than 1, the sequence exhibits exponential growth, while if $r$ is between 0 and 1, the sequence exhibits exponential decay.
3. Knowing the common ratio allows you to predict future terms in a geometric sequence by repeatedly multiplying the previous term by $r$.
4. The common ratio is a key characteristic that distinguishes geometric sequences from arithmetic sequences, which have a constant additive difference between terms.
5. Understanding the concept of common ratio is essential for analyzing patterns, making predictions, and solving problems involving geometric sequences and series.

Review Questions

• Explain how the common ratio relates to the behavior of a geometric sequence.
  • The common ratio $r$ determines the behavior of a geometric sequence. If $r$ is greater than 1, the sequence exhibits exponential growth, where each term is larger than the previous term by a factor of $r$. If $r$ is between 0 and 1, the sequence exhibits exponential decay, where each term is smaller than the previous term by a factor of $r$. Knowing the common ratio allows you to predict future terms in the sequence by repeatedly multiplying the previous term by $r$.
• Describe the difference between a geometric sequence and an arithmetic sequence in terms of the common ratio.
  • The key difference between a geometric sequence and an arithmetic sequence is the relationship between consecutive terms. In a geometric sequence, each term is obtained by multiplying the previous term by a constant, known as the common ratio $r$. In an arithmetic sequence, each term is obtained by adding a constant difference to the previous term. Unlike an arithmetic sequence, the common ratio is a crucial characteristic that defines the behavior of a geometric sequence and allows for the prediction of future terms.
• Analyze the importance of understanding the concept of common ratio in the context of geometric sequences and series.
  • Understanding the concept of common ratio is essential for analyzing, predicting, and solving problems involving geometric sequences and series. The common ratio $r$ determines the pattern of growth or decay in the sequence, allowing you to forecast future terms and calculate sums of infinite geometric series. Mastering the common ratio is crucial for recognizing the underlying structure of geometric sequences, applying appropriate formulas, and making connections between the sequence and its real-world applications, such as exponential growth and decay phenomena.

Related terms

Geometric Sequence: A sequence where each term is obtained by multiplying the previous term by a constant, known as the common ratio.

Arithmetic Sequence: A sequence where each term is obtained by adding a constant difference to the previous term, unlike a geometric sequence which uses a constant ratio.

Exponential Growth: A pattern of growth where each term is a fixed multiple of the previous term, characteristic of geometric sequences with a common ratio greater than 1.