Geometric Sequences and Series
- Arithmetic sequences differ by a common difference $d$ each term.
- Geometric sequences are similar, but differ by a common ratio $r$ each term instead, where the next term is the previous term multiplied by $r$.
Definition and Basic Properties
- A geometric sequence is defined by its first term $u_1$ and common ratio $r$.
- The general form of a geometric sequence is: $$u_1, u_1r, u_1r^2, u_1r^3, ..., u_1r^{n-1}$$ where $n$ is the position of the term in the sequence.
- If we have a sequence 2, 6, 18, 54, ..., we can identify that:
- The first term $a = 2$
- The common ratio $r = \frac{6}{2} = 3$
- This sequence can be written as $2, 2(3), 2(3^2), 2(3^3), ...$
- The common ratio of a geometric sequence can be between 0 and 1, in which case the terms in the sequence decrease.
- For example, the first five terms of a geometric series with $u_1 = 8$ and $r = \frac12$ would be $8, 4, 2, 1, \frac12$.
- The common ratio can also be negative, in which case the terms alternate sign.
- For example, the first five terms of a geometric series with $u_1 = 2$ and $r = -3$ would be $2, -6, 18, -54, 162$.
The nth Term Formula
The formula for the nth term of a geometric sequence is: $$u_n = u_1r^{n-1}$$ where
- $a_n$ is the nth term
- $u_1$ is the first term
- $r$ is the common ratio
- $n$ is the position of the term
- To find the common ratio $r$, divide any term by the previous term in the sequence.
- This should give the same result for any pair of consecutive terms.
