Geometric sequence: Difference between revisions

Revision as of 09:03, 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

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

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

Infinite Geometric Sequences

An infinite geometric sequence is a geometric sequence with an infinite number of terms. If the common ratio is small, the sum of the terms will approach a fixed limit. In this case, "small" means $|r|<1$ . We say that the sum of the terms of this sequence is a convergent sum.

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

$S = \frac{a_0}{1-r}$

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

One common instance of summing infinite geometric sequences is the decimal expansion of most rational numbers. For instance, $0.33333\ldots = \frac 3{10} + \frac3{100} + \frac3{1000} + \frac3{10000} + \ldots$ has first term $a_0 = \frac 3{10}$ and common ratio $\frac1{10}$ , so the infinite sum has value $S = \frac{\frac3{10}}{1-\frac1{10}} = \frac13$ , just as we would have expected.

@@ Line 17: / Line 17: @@
 where <math>a_0</math> is the first term in the sequence, and <math>r</math> is the common ratio.
-==Infinate Geometric Sequences==
+==Infinite 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 <math>|r|<1</math>.
+An infinite geometric sequence is a geometric sequence with an infinite number of terms.  If the common ratio is small, the sum of the terms will approach a fixed [[limit]].  In this case, "small" means <math>|r|<1</math>.  We say that the sum of the terms of this sequence is a [[convergent sum]].
-For instance, the series <math>1 + \frac12 + \frac14 + \frac18 + \cdots</math>, sums to 2.  The general fromula for the sum of such a sequence is:
+For instance, the series <math>1 + \frac12 + \frac14 + \frac18 + \cdots</math>, sums to 2.  The general formula for the sum of such a sequence is:
-<math>S = \frac{a_1}{1-r}</math>
+<math>S = \frac{a_0}{1-r}</math>
-Again, <math>a_1</math> is the first term in the sequence, and <math>r</math> is the common ratio.
+Again, <math>a_0</math> is the first term in the sequence, and <math>r</math> is the common ratio.
+One common instance of summing infinite geometric sequences is the [[decimal expansion]] of most [[rational number]]s.  For instance, <math>0.33333\ldots = \frac 3{10} + \frac3{100} + \frac3{1000} + \frac3{10000} + \ldots</math> has first term <math>a_0 = \frac 3{10}</math> and common ratio <math>\frac1{10}</math>, so the infinite sum has value <math>S = \frac{\frac3{10}}{1-\frac1{10}} = \frac13</math>, just as we would have expected.
 ==See Also==
 [[arithmetic sequence|Arithmetic Sequences]]

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Geometric sequence: Difference between revisions

Revision as of 09:03, 23 June 2006

Contents

Definition

Summing a Geometric Sequence

Infinite Geometric Sequences

See Also