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Geometric progression

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

$ar^0=a,ar^1=ar,ar^2,ar^3,...\,$

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, ....

A non-zero geometric progression shows exponential growth or exponential decay.

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.

Geometric series

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

$a(r^0+r^1+r^2+r^3+...+r^{n-1})\,$

Multiplying $(1+r+r^2+r^3+...+r^{n-1})\,$ by $(1-r)\,$ equals $(1-r^n)\,$ since all the other terms cancel in pairs.

Rearranging gives the convenient formula for a geometric series:

$a\sum_{k=0}^{n-1} r^k=a\frac{r^{n}-1}{r-1}$

An interesting relationship for a geometric series is given by:

$a(r^0+r^1+r^2+r^3+r^4+r^5+r^6+r^7+r^8+...) = a(r^0+r^1+r^2)(r^0+r^3+r^6+...)\,$

For example,

(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 × 2^x 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

$\sum_{k=m}^n x^k=\frac{x^{n+1}-x^m}{x-1}$

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

$\sum_{k=0}^\infty x^k=\frac{1}{1-x}$

which is valid whenever |x| < 1; it is a consequence of the above formula for finite geometric series by taking the limit for n→∞.

This last formula is actually valid in every Banach algebra, as long as the norm of x is less than one, and also in the field of p-adic numbers if |x|_p < 1.

Also useful is the formula

$\sum_{k=0}^\infty k\cdot x^k=\frac{x}{(1-x)^2}$

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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