Common Ratio

Definition

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

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 calculated as $a \cdot r^{n-1}$.
3. For a convergent infinite geometric series with common ratio $r$, the sum of the series is given by the formula $\frac{a}{1-r}$, where $a$ is the first term.
4. The common ratio determines whether a geometric sequence is increasing ($r > 1$), decreasing ($0 < r < 1$), or constant ($r = 1$).
5. The common ratio is a key characteristic that distinguishes geometric sequences from arithmetic sequences, which have a constant difference between terms instead of a common ratio.

Review Questions

• Explain how the common ratio is used to generate terms in a geometric sequence.
  • The common ratio, denoted as $r$, is the constant multiplicative factor that is used to generate each new term in a geometric sequence from the previous term. If the first term of the sequence is $a$, then the $n^{th}$ term can be calculated as $a \cdot r^{n-1}$, where $n$ is the position of the term in the sequence. The common ratio $r$ determines whether the sequence is increasing ($r > 1$), decreasing ($0 < r < 1$), or constant ($r = 1$).
• Describe the relationship between the common ratio and the sum of an infinite geometric series.
  • For a convergent infinite geometric series with common ratio $r$, the sum of the series can be calculated using the formula $\frac{a}{1-r}$, where $a$ is the first term of the series. The common ratio $r$ is a critical factor in determining the convergence and sum of the series. If $|r| < 1$, the series converges, and the sum can be found using the formula. If $|r| \geq 1$, the series diverges, and the sum does not exist.
• Analyze how the common ratio distinguishes geometric sequences from arithmetic sequences, and explain the implications of this difference.
  • The key distinction between geometric and arithmetic sequences is the method used to generate new terms. In a geometric sequence, each term is obtained by multiplying the previous term by a constant common ratio $r$. In an arithmetic sequence, each term is obtained by adding a constant difference to the previous term. This difference in the underlying pattern has important implications. The common ratio determines whether a sequence is increasing, decreasing, or constant, while the constant difference in an arithmetic sequence results in a linear growth pattern. Additionally, the common ratio is a crucial factor in the formulas used to calculate the $n^{th}$ term and the sum of an infinite series for geometric sequences.

Related terms

Geometric Sequence:

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

Arithmetic Sequence:

An arithmetic sequence is a sequence where each term is obtained by adding a constant difference to the previous term, unlike a geometric sequence which uses a common ratio.

Infinite Geometric Series:

An infinite geometric series is a series where the terms form a geometric sequence, and the sum of the infinite series can be calculated using the common ratio.