Science Fair Project Encyclopedia
In mathematics, a geometric progression (also inaccurately known as a geometric series, see below) is a sequence of numbers such that the quotient of any two successive members of the sequence is a constant called the common ratio of the sequence.
Thus without loss of generality a geometric sequence can be written as
where r ≠ 0 is the common ratio and a is a scale factor. Thus the common ratio gives a family of geometric sequences whose starting value is determined by the scale factor. Pedantically speaking, the case r = 0 ought to be excluded, since the common ratio is not even defined; but the sequence that is always 0 is included, by convention.
For example, a sequence with a common ratio of 2 and a scale factor of 1 is
- 1, 2, 4, 8, 16, 32, ....
and a sequence with a common ratio of 2/3 and a scale factor of 729 is
- 729 (1, 2/3, 4/9, 8/27, 16/81, 32/243, 64/729, ....) = 729, 486, 324, 216, 144, 96, 64, ....
and finally a sequence with a common ratio of −1 and a scale factor of 3 is
- 3 (1, −1, 1, −1, 1, −1, 1, −1, 1, −1, ....) = 3, −3, 3, −3, 3, −3, 3, −3, 3, −3, ....
Compare this with an arithmetic progression showing linear growth (or decline) such as 4, 15, 26, 37, 48, .... Note that the two kinds of progression are related: taking the logarithm of each term in a geometric progression yields an arithmetic one.
A geometric series is, strictly speaking, the sum of the numbers in a geometric progression. Thus the geometric series for the n terms of a geometric progression is
Multiplying by equals since all the other terms cancel in pairs.
Rearranging gives the convenient formula for a geometric series:
An interesting relationship for a geometric series is given by:
- (1 + 2 + 4 + 8 + 16 + 32 + 64 + 128 + 256 +...)
can be written as
- (1 + 2 + 4)(1 + 8 + 64 +...)
Since a geometric series is a sum of terms in which two successive terms always have the same ratio,
- 4 + 8 + 16 + 32 + 64 + 128 + 256 + ...
is a geometric series with a common ratio of 2. This is the same as 2 × 2x where x increases by one for each number. It is called a geometric series because it occurs when comparing the length, area, volume, etc. of a shape in different dimensions.
The sum of a geometric series whose first term is a power of the common ratio can be computed quickly with the formula
which is valid for all natural numbers m ≤ n and all numbers x≠ 1 (or more generally, for all elements x in a ring such that x − 1 is invertible). This formula can be verified by multiplying both sides with x - 1 and simplifying.
Using the formula, we can determine the above sum: (29 − 22)/(2 − 1) = 508. The formula is also extremely useful in calculating annuities: suppose you put $2,000 in the bank every year, and the money earns interest at an annual rate of 5%. How much money do you have after 6 years?
- 2,000 · 1.056 + 2,000 · 1.055 + 2,000 · 1.054 + 2,000 · 1.053 + 2,000 · 1.052 + 2,000 · 1.051
- = 2,000 · (1.057 − 1.05)/(1.05 − 1)
- = 14,284.02
An infinite geometric series is an infinite series whose successive terms have a common ratio. Such a series converges if and only if the absolute value of the common ratio is less than one; its value can then be computed with the formula
which is valid whenever |x| < 1; it is a consequence of the above formula for finite geometric series by taking the limit for n→∞.
Also useful is the formula
which can be seen as x times the derivative of the infinite geometric series. This formula only works for |x| < 1, as well.
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