Difference between revisions of "Geometric sequence"

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(Definition)
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==Definition==
 
==Definition==
  
A geometric sequence is a sequence of numbers where the nth term of the sequence is a 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.
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A geometric sequence is a sequence of numbers where the nth term of the sequence is a 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 as:
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<math>a_n = r*a_{n-1}, n \geq 2</math>
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with a fixed <math>a_1</math> and common ratio <math>r</math>.  Using this definition, the <math>n</math>th term may be found explicityly with:
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<math>\displaystyle a_n = a_1*r^{n-1}</math>
  
 
==Summing a Geometric Sequence==
 
==Summing a Geometric Sequence==

Revision as of 02:54, 23 June 2006

Definition

A geometric sequence is a sequence of numbers where the nth term of the sequence is a 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 as:

$a_n = r*a_{n-1}, n \geq 2$

with a fixed $a_1$ and common ratio $r$. Using this definition, the $n$th term may be found explicityly with:

$\displaystyle a_n = a_1*r^{n-1}$

Summing a Geometric Sequence

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

$S_n = \frac{a_1(r^{n+1}-1)}{r-1}$

where $a_1$ 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 $1 + \frac12 + \frac14 + \frac18 + ...$, 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.

See Also

Arithmetic Sequences