Difference between revisions of "Geometric sequence"

Revision as of 08:54, 23 June 2006

Definition

A geometric sequence is a sequence of numbers in which each term is a fixed multiple of the previous term. For example: 1, 2, 4, 8, 16, 32, ... is a geometric sequence because each term is twice the previous term. In this case, 2 is called the common ratio of the sequence. More formally, a geometric sequence may be defined recursively by:

$a_n = r\cdot a_{n-1}, n \geq 1$

with a fixed $a_0$ and common ratio $r$ . Using this definition, the $n$ th term has the closed-form:

$\displaystyle a_n = a_0\cdot r^n$

Summing a Geometric Sequence

The sum of the first $n$ terms of a geometric sequence is given by

$S_n = a_0 + a_1 + \ldots + a_{n - 1} = a_0\cdot\frac{r^n-)}{r-1}$

where $a_0$ is the first term in the sequence, and $r$ is the common ratio.

Infinate Geometric Sequences

An infinate geometric sequence is a geometric sequence with an infinate number of terms. These sequences can have sums, sometimes called limits, if $|r|<1$ .

For instance, the series $1 + \frac12 + \frac14 + \frac18 + \cdots$ , sums to 2. The general fromula for the sum of such a sequence is:

$S = \frac{a_1}{1-r}$

Again, $a_1$ is the first term in the sequence, and $r$ is the common ratio.

@@ Line 13: / Line 13: @@
 The sum of the first <math>n</math> terms of a geometric sequence is given by
-<math>S_n = \frac{a_1(r^{n+1}-1)}{r-1}</math>
+<math>S_n = a_0 + a_1 + \ldots + a_{n - 1} = a_0\cdot\frac{r^n-)}{r-1}</math>
-where <math>a_1</math> is the first term in the sequence, and <math>r</math> is the common ratio.
+where <math>a_0</math> is the first term in the sequence, and <math>r</math> is the common ratio.
 ==Infinate Geometric Sequences==

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Difference between revisions of "Geometric sequence"

Revision as of 08:54, 23 June 2006

Contents

Definition

Summing a Geometric Sequence

Infinate Geometric Sequences

See Also