Geometric sequence

Definition

A geometric sequence is a sequence of numbers where each term is found by multiplying the previous term by a constant called the common ratio. The general form of a geometric sequence is $a, ar, ar^2, ar^3,$ and so on.

5 Must Know Facts For Your Next Test

1. A geometric sequence can be finite or infinite depending on the number of terms.
2. The common ratio ($r$) can be positive, negative, or even a fraction.
3. The $n$-th term of a geometric sequence can be expressed as $a_n = ar^{n-1}$ where $a$ is the first term.
4. The sum of the first $n$ terms (finite series) of a geometric sequence is given by $S_n = \frac{a(1-r^n)}{1-r}$ if $r \neq 1$.
5. For an infinite geometric series with $|r| < 1$, the sum converges to $\frac{a}{1-r}$.

Review Questions

• What is the common ratio in the geometric sequence 3, 6, 12, 24?
• How do you find the sum of the first 5 terms in a geometric sequence with $a = 2$ and $r = 3$?
• Does the infinite geometric series with $a = 4$ and $r = -0.5$ converge? If yes, what is its sum?

Related terms

Arithmetic Sequence:

A sequence of numbers where each term after the first is found by adding a constant difference to the previous term.

Series:

The sum of the terms in a sequence.

Convergence:

A property indicating that as more terms are added in a series or as n approaches infinity in sequences, it approaches a specific value.