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:
![$a_n = r\cdot a_{n-1}, n > 1$](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:
![$\displaystyle a_n = a_1\cdot r^{n-1}$](http://latex.artofproblemsolving.com/6/a/f/6af6c46ab81c06ca7cc7d8a5a48958bf74fe46dc.png)
Summing a Geometric Sequence
The sum of the first terms of a geometric sequence is given by
![$S_n = a_1 + a_2 + \ldots + a_n = a_1\cdot\frac{r^n-1}{r-1}$](http://latex.artofproblemsolving.com/c/4/1/c41e3dd3b3d2b52684fe34ff2200a63f0ca038e0.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:
![$S = \frac{a_1}{1-r}$](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/1/9/4/194783d69c4452bb4fa3cb1f64df3ee1c42373b6.png)
Multiplying by yields,
![$S \cdot r=a_1r+a_1r^2+a_1r^3+\cdots$](http://latex.artofproblemsolving.com/2/5/3/253e6784d10fe90d620d9f1db5bb0b5f68957c2e.png)
We subtract these two equations to obtain:
![$S-Sr=a_1$](http://latex.artofproblemsolving.com/6/c/c/6cc6a7d3c64f9e92e549a690649ee9f26ebaf005.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
![$\displaystyle S(1-r)=a_1$](http://latex.artofproblemsolving.com/d/c/d/dcd9f6938cad673c56e8267e5b6bd24fc35842d5.png)
thus,
![$S=\frac{a_1}{1-r}$](http://latex.artofproblemsolving.com/d/a/4/da4b5ab6da1c561dfde66ac6a1690b967bc53319.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.