Difference between revisions of "Geometric sequence"
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− | + | In [[algebra]], a '''geometric sequence''', sometimes called a '''geometric progression''', is a [[sequence]] of numbers such that the ratio between any two consecutive terms is constant. This constant is called the '''common difference''' of the sequence. | |
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+ | For example, <math>7, 14, 28, 56</math> is a geometric sequence with common ratio <math>2</math> and <math>100, -50, 25, -25/2, \ldots</math> is a geometric sequence with common ratio <math>-1/2</math>; however, <math>1, 3, 9, 19</math> and <math>-3, 1, 5, 9</math> are not geometric sequences, as the ratio between consecutive terms varies. | ||
+ | |||
+ | More formally, the sequence <math>a_1, a_2, \ldots , a_n</math> is a geometric progression if and only if <math>\frac{a_2}{a_1} = \frac{a_3}{a_2} = \cdots = \frac{a_n}{a_{n-1}}</math>. This definition appears most frequently in its three-term form; that constants <math>a</math>, <math>b</math>, and <math>c</math> are in arithmetic progression if and only if <math>b - a = c - b</math>. | ||
<center><math>a_n = r\cdot a_{n-1}, n > 1</math></center> | <center><math>a_n = r\cdot a_{n-1}, n > 1</math></center> | ||
with a fixed first term <math>a_1</math> and common ratio <math>r</math>. Using this definition, the <math>n</math>th term has the closed-form: | with a fixed first term <math>a_1</math> and common ratio <math>r</math>. Using this definition, the <math>n</math>th term has the closed-form: | ||
+ | |||
+ | <center><math>a_n = a_1\cdot r^{n-1}</math></center> | ||
+ | |||
+ | ==Properties== | ||
+ | The <math>n</math>th term has the closed-form: | ||
<center><math>a_n = a_1\cdot r^{n-1}</math></center> | <center><math>a_n = a_1\cdot r^{n-1}</math></center> | ||
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*[[Sequence]] | *[[Sequence]] | ||
*[[Series]] | *[[Series]] | ||
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+ | [[Category:Algebra]] [[Category:Sequences and series]] [[Category:Definition]] |
Revision as of 15:04, 3 November 2021
In algebra, a geometric sequence, sometimes called a geometric progression, is a sequence of numbers such that the ratio between any two consecutive terms is constant. This constant is called the common difference of the sequence.
For example, is a geometric sequence with common ratio
and
is a geometric sequence with common ratio
; however,
and
are not geometric sequences, as the ratio between consecutive terms varies.
More formally, the sequence is a geometric progression if and only if
. This definition appears most frequently in its three-term form; that constants
,
, and
are in arithmetic progression if and only if
.
![$a_n = r\cdot a_{n-1}, n > 1$](http://latex.artofproblemsolving.com/6/3/7/6371ffa98726d4d83a638615d4e618e35457c58d.png)
with a fixed first term and common ratio
. Using this definition, the
th term has the closed-form:
![$a_n = a_1\cdot r^{n-1}$](http://latex.artofproblemsolving.com/f/a/7/fa731a133dc037e4eece365b97e9c7d180a55c44.png)
Contents
Properties
The th term has the closed-form:
![$a_n = a_1\cdot r^{n-1}$](http://latex.artofproblemsolving.com/f/a/7/fa731a133dc037e4eece365b97e9c7d180a55c44.png)
Summing a Geometric Sequence
The sum of the first terms of a geometric sequence is given by
![$S_n = a_1 + a_2 + \cdots + a_n = a_1\cdot\frac{r^n-1}{r-1}$](http://latex.artofproblemsolving.com/9/a/2/9a2644b40fca72573d406c700b17623e8055f2ad.png)
where is the first term in the sequence, and
is the common ratio.
Proof
The geometric sequence can be rewritten as where
is the number of terms,
is the common ratio, and
is the first term. Multiplying by
yields
so
.
Infinite Geometric Sequences
An infinite geometric sequence is a geometric sequence with an infinite number of terms. If the common ratio is small, the terms will approach 0 and the sum of the terms will approach a fixed limit. In this case, "small" means . We say that the sum of the terms of this sequence is a convergent sum.
For instance, the series , sums to 2. The general formula for the sum of such a sequence is:
![$S = \frac{a_1}{1-r}$](http://latex.artofproblemsolving.com/0/4/6/04695885b4e05009137c6d4ff379ed3af18c7e38.png)
Where is the first term in the sequence, and
is the common ratio.
Proof
Let the sequence be
![$S=a_1+a_1r+a_1r^2+a_1r^3+\cdots.$](http://latex.artofproblemsolving.com/b/6/d/b6d677925be08b02d6da3630b7c26f75f85f4156.png)
Multiplying by yields,
![$S \cdot r=a_1r+a_1r^2+a_1r^3+\cdots.$](http://latex.artofproblemsolving.com/0/b/7/0b725c4e6ccf1e82278a44e2a9a4c74e6890826f.png)
We subtract these two equations to obtain:
![$S-Sr=a_1.$](http://latex.artofproblemsolving.com/9/c/4/9c46726a6936bc9ab959e769d65ee6a0619c0c1a.png)
There is only one term on the right side of the equation because the rest of the terms cancel out after subtraction. Finally, we can factor and divide to get
![$S(1-r)=a_1$](http://latex.artofproblemsolving.com/1/e/0/1e0f0f05c594d4e178bf2d3f814f632f7913a842.png)
thus,
![$S=\frac{a_1}{1-r}.$](http://latex.artofproblemsolving.com/3/7/7/377efb8a7393c5eb8f07016e268ec9f005b70182.png)
This method of multiplying the sequence and subtracting equations, called telescoping, is a frequently used method to evaluate infinite sequences. In fact, the same method can be used to calculate the sum of a finite geometric sequence (given above).
Common uses
One common instance of summing infinite geometric sequences is the decimal expansion of most rational numbers. For instance, has first term
and common ratio
, so the infinite sum has value
, just as we would have expected.