Difference between revisions of "Geometric sequence"
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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). | 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 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. | 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. |
Revision as of 11:50, 19 December 2018
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:
![$a_n = r\cdot a_{n-1}, n > 1$](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
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 amount of terms,
is the common ratio, and
is the first term. Multiplying in
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 RHS 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.