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.
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The common ratio, denoted as $r$, is the ratio between any two consecutive terms in a geometric sequence.
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 $a \cdot r^{n-1}$.
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.
The common ratio determines whether a geometric sequence is increasing ($r > 1$), decreasing ($0 < r < 1$), or constant ($r = 1$).
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.
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.
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.