Geometric sequence
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:
![$a_n = r\cdot a_{n-1}, n \geq 1$](http://latex.artofproblemsolving.com/d/f/9/df9ebc4a788f5edce468317c62d193fb135230d6.png)
with a fixed and common ratio
. Using this definition, the
th term has the closed-form:
![$\displaystyle a_n = a_0\cdot r^n$](http://latex.artofproblemsolving.com/1/8/a/18a3661fc1e968b22abd63f188574bd96551977e.png)
Summing a Geometric Sequence
The sum of the first terms of a geometric sequence is given by
![$S_n = a_0 + a_1 + \ldots + a_{n - 1} = a_0\cdot\frac{r^n-1}{r-1}$](http://latex.artofproblemsolving.com/1/5/a/15a3fa5c5f70b5749e37d211e01ae7ccfddc36a0.png)
where is the first term in the sequence, and
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 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:
![$S = \frac{a_0}{1-r}$](http://latex.artofproblemsolving.com/5/c/5/5c52aa7d0321d4e107063cc911c4585eb1e820a0.png)
Where is the first term in the sequence, and
is the common ratio.
"Proof": Let the sequence be
![$S=a_0+a_0r+a_0r^2+a_0r^3+\cdots$](http://latex.artofproblemsolving.com/0/4/b/04bd4e810ebc042658c0680f8e4ba99893096871.png)
Multiplying by yields,
![$S \cdot r=a_0r+a_0r^2+a_0r^3+\cdots$](http://latex.artofproblemsolving.com/7/a/1/7a174c2db03739e780169ca616061d664e065375.png)
We subtract these two equations to obtain:
![$S-S\cdot r=a_0$](http://latex.artofproblemsolving.com/a/c/c/acc512a63eec4f19864853375399ae44982eb9c8.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
![$\displaystyle S(1-r)=a_0$](http://latex.artofproblemsolving.com/5/c/f/5cfcf547690e553a8452fe9a23fa3b92f60ba78f.png)
thus,
![$S=\frac{a_0}{1-r}$](http://latex.artofproblemsolving.com/b/8/2/b82633ec4030dbfef40c7d030e7c03090e2e2dde.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).
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.