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. The general form of a geometric sequence can be written as $a, ar, ar^2, ar^3, \ldots$.
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The common ratio ($r$) in a geometric sequence is found by dividing any term by its preceding term.
The $n$-th term of a geometric sequence can be calculated using the formula $a_n = a \cdot r^{n-1}$, where $a$ is the first term and $n$ is the position of the term in the sequence.
The sum of the first $n$ terms of a geometric series (finite series) can be calculated using the formula $S_n = a \frac{1-r^n}{1-r}$ if $r \neq 1$.
For an infinite geometric series with $|r| < 1$, the sum converges to $\frac{a}{1-r}$.
Geometric sequences can model exponential growth or decay processes such as population growth, radioactive decay, and interest calculations.
Review Questions
How do you find the common ratio in a geometric sequence?
What is the formula for finding the sum of the first $n$ terms in a finite geometric series?
What condition must be met for an infinite geometric series to converge?