geometric sequence
(another Prime Pages' Glossary entries)
The Prime Glossary
Glossary: Prime Pages: Top 5000: A geometric sequence is a sequence (finite or infinite) of real numbers for which each term is the previous term multiplied by a constant (called the common ratio).  For example, starting with 3 and using a common ratio of 2 we get the finite geometric sequence: 3, 6, 12, 24, 48; and also the infinite sequence
3, 6, 12, 24, 48, ..., 3.2n ...
In general, the terms of a geometric sequence have the form an = a.rn (n=0,1,2,...) for fixed numbers a and r.

When we add the terms of a geometric sequence, we get a geometric series.  If it is a finite series, then we add its terms to get the series' sum

a + a.r + a.r2 + ... + a.rn = (a-a.rn+1)/(1-r)
When |r| < 1, then we also can sum the infinite series, and it will have the sum a/(1-r). (When |r| > 1, then the series diverges and has no sum.)

See Also: ArithmeticSequence


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