A geometric sequence is a sequence of numbers in which each term is obtained by multiplying the previous term by a constant factor. That constant factor is the common ratio, and it creates an exponential growth or decay pattern.
Definition A sequence \(\{a_n\}\) is a geometric sequence if there exists a constant \(r\) such that \[ \frac{a_{n+1}}{a_n}=r \quad \text{for all } n \text{ where } a_n\ne 0. \] Equivalently, the recursive rule is \(a_{n+1}=r\cdot a_n\).
How to find the common ratio
Given consecutive terms \(a_n\) and \(a_{n+1}\), the common ratio is computed by:
\[ r=\frac{a_{n+1}}{a_n}\quad (a_n\ne 0). \]
If the ratio is not the same for different consecutive pairs, the sequence is not geometric.
Formula for the nth term
If the first term is \(a_1\) and the common ratio is \(r\), then the nth term of a geometric sequence is:
\[ a_n=a_1\cdot r^{\,n-1}. \]
Example: compute a term
Suppose \(a_1=3\) and \(r=2\). Then:
\[ a_5 = 3\cdot 2^{4}=3\cdot 16=48. \]
Geometric series and the partial sum
The sum of the first \(n\) terms of a geometric sequence, \[ S_n=a_1+a_2+\cdots+a_n, \] is called a geometric series.
Partial sum For \(r\ne 1\), \[ S_n=a_1\cdot\frac{1-r^n}{1-r}. \] If \(r=1\), then every term equals \(a_1\) and \(S_n=n\cdot a_1\).
Example: compute a partial sum
For \(a_1=3\), \(r=2\), and \(n=5\),
\[ S_5=3\cdot\frac{1-2^5}{1-2} =3\cdot\frac{1-32}{-1} =3\cdot 31 =93. \]
This matches \(3+6+12+24+48=93\).
Recognizing a geometric sequence from a list of terms
Consider the sequence \(81, 27, 9, 3, 1, \dots\). The ratio of consecutive terms is:
\[ \frac{27}{81}=\frac{1}{3},\quad \frac{9}{27}=\frac{1}{3},\quad \frac{3}{9}=\frac{1}{3}. \]
Since the ratio is constant, it is a geometric sequence with \(a_1=81\) and \(r=\frac{1}{3}\).
Visualization: terms of a geometric sequence along an index axis
The plot below shows the first six terms of the geometric sequence with \(a_1=3\) and \(r=2\). Each step in \(n\) doubles the value, illustrating exponential growth.
Summary
A geometric sequence is defined by a constant common ratio \(r\). With first term \(a_1\), the nth term is \(a_n=a_1\cdot r^{n-1}\), and the partial sum of the first \(n\) terms is \(S_n=a_1\cdot\frac{1-r^n}{1-r}\) for \(r\ne 1\).