Common Ratio

Definition

The common ratio is a constant ratio between consecutive terms in a geometric sequence. It represents the multiplicative factor that is used to generate each successive term from the previous one.

5 Must Know Facts For Your Next Test

1. The common ratio, denoted as $r$, is the ratio between any two consecutive terms in a geometric sequence.
2. If the first term of a geometric sequence is $a$ and the common ratio is $r$, then the $n$th term can be expressed as $a \cdot r^{n-1}$.
3. A geometric sequence converges if the common ratio $|r| < 1$, and diverges if $|r| > 1$.
4. The sum of the first $n$ terms of a geometric sequence with common ratio $r$ is given by the formula $S_n = a \cdot \frac{1 - r^n}{1 - r}$.
5. Geometric sequences are often used to model exponential growth and decay processes in various fields, such as finance, biology, and physics.

Review Questions

• Explain the relationship between the common ratio and the general term of a geometric sequence.
  • The common ratio, denoted as $r$, is the constant ratio between any two consecutive terms in a geometric sequence. If the first term of the sequence is $a$, then the $n$th term can be expressed as $a \cdot r^{n-1}$. This means that each term is obtained by multiplying the previous term by the common ratio $r$. The common ratio is the key factor that determines the pattern and behavior of a geometric sequence.
• Describe the conditions for the convergence and divergence of a geometric sequence based on the common ratio.
  • The behavior of a geometric sequence is determined by the value of the common ratio $r$. If $|r| < 1$, the sequence converges, meaning the terms get closer and closer to 0 as $n$ increases. If $|r| > 1$, the sequence diverges, with the terms growing larger and larger without bound. When $|r| = 1$, the sequence is neither converging nor diverging, but rather oscillating between constant values. The common ratio is a crucial factor in understanding the long-term behavior of a geometric sequence.
• Explain how the common ratio is used to derive the formula for the sum of the first $n$ terms of a geometric sequence.
  • The formula for the sum of the first $n$ terms of a geometric sequence with common ratio $r$ is given by $S_n = a \cdot \frac{1 - r^n}{1 - r}$, where $a$ is the first term. This formula is derived by recognizing the pattern in the sequence and exploiting the relationship between consecutive terms due to the common ratio $r$. The common ratio allows us to express each term in the sequence in terms of the first term and the common ratio, leading to the closed-form expression for the sum of the first $n$ terms.

Related terms

Geometric Sequence: A sequence where each term is obtained by multiplying the previous term by a constant ratio, 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 Function: A function that grows or decays at a rate proportional to its current value, which is a characteristic of geometric sequences with a common ratio.