Lesson 3: Analyzing Geometric Sequences and Series

Property

A geometric sequence is a sequence where the ratio between consecutive terms is always the same.

The ratio between consecutive terms, $\frac{a_n}{a_{n-1}}$, is $r$, the common ratio. $n$ is greater than or equal to two.

Examples

• The sequence $5, 15, 45, 135, \ldots$ is geometric because the ratio between consecutive terms is always 3. The common ratio is $r=3$.

Property

A geometric sequence can be written as an exponential function: $a_n = a_1 \cdot r^{n-1}$ where $a_1$ is the initial value and $r$ is the base of the exponential expression.

Examples

Property

The general term of a geometric sequence with first term $a_1$ and the common ratio $r$ is

Examples

• To find the 10th term of a sequence where $a_1 = 4$ and $r = 2$, we use the formula: $a_{10} = 4 \cdot 2^{10-1} = 4 \cdot 2^9 = 4 \cdot 512 = 2048$.