Geometric progression

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In mathematics, a geometric progression (also known as a geometric sequence, and, inaccurately, 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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_1=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.

Formulae

Progressions allow the use of a few simple formulae to find each term. The nth term can be defined as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_n = a\,r^{n-1} \quad \mbox{where n is an integer such that }n \ge 1}

The common ratio is then

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=\left(\frac{a_n}{a}\right)^{1/(n-1)} \quad \mbox{where n is an integer such that }n \ge 2}

and the scale factor is

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a=\frac{a_n}{r^{n-1}}.}

Examples

A sequence with a common ratio of 2 and a scale factor of 1 is

1, 2, 4, 8, 16, 32, ....

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, ....

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.

If the common ratio is:

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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{k=0}^{n} x^k = x^0+x^1+x^2+x^3+...+x^n \,}

We can find a simpler formula for this sum by multiplying both sides of the above equation by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (1-x)\,} and we'll see that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (1-x) \sum_{k=0}^{n} x^k = 1-x^{n+1}\,}

since all the other terms cancel. Rearranging gives the convenient formula for a geometric series:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{k=0}^{n} x^k = \frac{1-x^{n+1}}{1-x}}

Note: If one were to begin the sum not from 0, but from a higher term, say m, then

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{k=m}^n x^k=\frac{x^m-x^{n+1}}{1-x}}

Differentiating the sum with respect to x allows us to arrive at formulae for sums of the form

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{k=0}^n k^s x^k}

For example:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{d}{dx}\sum_{k=0}^nx^k = \sum_{k=0}^nkx^{k-1}= \frac{1-x^{n+1}}{(1-x)^2}-\frac{(n+1)x^n}{1-x}}


Infinite geometric series

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 from the finite sum formulae by setting terms containing Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^n} to zero:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{k=0}^\infty x^k=\frac{1}{1-x}}

or, in cases where the sum does not start at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k=0} ,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{k=a}^\infty x^k=\frac{x^a}{1-x}}

Both are valid only for |x| < 1. 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. As in the case for a finite sum, we can differentiate to calculate formulae for related sums. For example:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{d}{dx}\sum_{k=0}^\infty x^k = \sum_{k=0}^\infty kx^{k-1}= \frac{1}{(1-x)^2}}

This formula only works for |x| < 1, as well.

See also

ca:Progressió geomètrica cs:Geometrická posloupnost de:Geometrische Reihe fr:Série géométrique id:Deret ukur it:Progressione geometrica he:סדרה הנדסית ko:등비수열 lt:Geometrinė progresija nl:Meetkundige rij pl:Szereg geometryczny pt:Progressão geométrica ru:Геометрическая прогрессия sv:Geometrisk funktion uk:Геометрична прогресія zh:等比数列