Geometric sequence

Definition

A geometric sequence is 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. It can be expressed as $a, ar, ar^2, ar^3, \ldots$ where $a$ is the first term and $r$ is the common ratio.

5 Must Know Facts For Your Next Test

1. The formula for the nth term of a geometric sequence is $a_n = ar^{n-1}$.
2. The sum of the first n terms (finite sum) of a geometric sequence is given by $S_n = a \frac{1-r^n}{1-r}$ if $r \ne 1$.
3. For an infinite geometric series with $|r| < 1$, the sum converges to $S = \frac{a}{1-r}$.
4. If $r > 1$, the terms in the sequence grow exponentially; if $0 < r < 1$, they decay exponentially.
5. Geometric sequences are often used in real-world applications such as calculating interest rates and population growth.

Review Questions

• What is the common ratio in a geometric sequence?
• How do you find the nth term of a geometric sequence?
• What condition must be met for an infinite geometric series to converge?

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 of a sequence.

Common Ratio: The fixed, non-zero number multiplied to each term to get to the next term in a geometric sequence.