Geometric sequence
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:
with a fixed and common ratio . Using this definition, the th term may be found explicityly with:
Summing a Geometric Sequence
The sum of the first terms of a geometric sequence is given by
where is the first term in the sequence, and 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 .
For instance, the series , sums to 2. The general fromula for the sum of such a sequence is:
Again, is the first term in the sequence, and is the common ratio.