Common ratio

Definition

In a geometric sequence, the common ratio is the constant factor between consecutive terms. It is usually denoted by $r$ and can be found by dividing any term by its preceding term.

5 Must Know Facts For Your Next Test

1. The common ratio $r$ in a geometric sequence can be calculated using $r = \frac{a_{n+1}}{a_n}$, where $a_n$ is the n-th term.
2. If the common ratio $|r| < 1$, the terms in the geometric sequence will get progressively smaller.
3. If $|r| > 1$, the terms in the geometric sequence will grow exponentially.
4. A geometric series with common ratio $r = 1$ has all terms equal, resulting in a constant sequence.
5. For an infinite geometric series to converge, the common ratio needs to satisfy $|r| < 1$.

Review Questions

• How do you calculate the common ratio of a geometric sequence?
• What happens to a geometric sequence when the common ratio is greater than one?
• Explain why an infinite geometric series converges for certain values of the common ratio.

Related terms

Geometric Sequence: A sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.

Geometric Series: The sum of the terms of a geometric sequence.

$n$-th Term Formula: $a_n = a_1 \cdot r^{(n-1)}$, where $a_1$ is the first term and $r$ is the common ratio.