Geometric progression

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

${\displaystyle a,ar,ar^{2},ar^{3},ar^{4},\ldots \,}$

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.

Formula

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

${\displaystyle a_{n}=a\,r^{n-1}}$	where n is an integer such that ${\displaystyle n\geq 1}$
The common ratio is then
${\displaystyle r=\left({\frac {a_{n}}{a}}\right)^{\frac {1}{n-1}}}$  or ${\displaystyle r={\sqrt[{n-1}]{\frac {a_{n}}{a}}}}$	where n is an integer such that ${\displaystyle n\geq 1}$

and the scale factor is

${\displaystyle a={\frac {a_{n}}{r^{n-1}}}.}$ 

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, ....

This sequence's behaviour depends on the value of the common ratio.

If the common ratio is:

• Positive, the terms will all be the same sign as the initial term.
• Negative, the terms will alternate between positive and negative.
• 0, the results will remain at zero.
• Greater than 1, there will be exponential growth towards infinity (positive).
• 1, the progression is a constant sequence.
• Between 1 and −1 but not zero, there will be exponential decay towards zero.
• −1, the progression is an alternating sequence (see alternating series)
• Less than −1, there will be exponential growth towards infinity (positive and negative).

A geometric progression with common ratio ${\displaystyle \notin \{0,\pm 1\}}$  shows exponential growth or exponential decay, as opposed to the linear growth (or decline) of an arithmetic progression such as 4, 15, 26, 37, 48, ....
This result was taken by T.R. Malthus as the mathematical foundation of his Principle of Population.

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 the sum of the numbers in a geometric progression:

${\displaystyle \sum _{k=0}^{n}ar^{k}=ar^{0}+ar^{1}+ar^{2}+ar^{3}+\cdots +ar^{n}\,}$ 

We can find a simpler formula for this sum by multiplying both sides of the above equation by ${\displaystyle (1-r)}$ , and we'll see that

${\displaystyle (1-r)\sum _{k=0}^{n}ar^{k}=a-ar^{n+1}\,}$ 

since all the other terms cancel. Rearranging (for ${\displaystyle r\neq 1}$ ) gives the convenient formula for a geometric series:

${\displaystyle \sum _{k=0}^{n}ar^{k}={\frac {a(1-r^{n+1})}{1-r}}}$ 

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

${\displaystyle \sum _{k=m}^{n}ar^{k}={\frac {a(r^{m}-r^{n+1})}{1-r}}}$ 

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

${\displaystyle \sum _{k=0}^{n}k^{s}r^{k}}$ 

For example:

${\displaystyle {\frac {d}{dr}}\sum _{k=0}^{n}r^{k}=\sum _{k=0}^{n}kr^{k-1}={\frac {1-r^{n+1}}{(1-r)^{2}}}-{\frac {(n+1)r^{n}}{1-r}}}$ 

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 ( | r | < 1 ). Its value can then be computed from the finite sum formulae

${\displaystyle \sum _{k=0}^{\infty }ar^{k}=\lim _{n\to \infty }{\sum _{k=0}^{n}ar^{k}}=\lim _{n\to \infty }{\frac {a(1-r^{n+1})}{1-r}}=\lim _{n\to \infty }{\frac {a}{1-r}}-\lim _{n\to \infty }{\frac {ar^{n+1}}{1-r}}={\frac {a}{1-r}}}$ 

For example,

${\displaystyle \sum _{k=0}^{\infty }(191)\left({\frac {6}{7}}\right)^{k}={\frac {191}{1-{\frac {6}{7}}}}=1337}$ 

In cases where the sum does not start at k = 0,

${\displaystyle \sum _{k=m}^{\infty }ar^{k}={\frac {ar^{m}}{1-r}}}$ 

Both formulae are valid only for | r | < 1. The latter formula is actually valid in every Banach algebra, as long as the norm of r is less than one, and also in the field of p-adic numbers if | r |_p < 1. As in the case for a finite sum, we can differentiate to calculate formulae for related sums. For example,

${\displaystyle {\frac {d}{dr}}\sum _{k=0}^{\infty }r^{k}=\sum _{k=0}^{\infty }kr^{k-1}={\frac {1}{(1-r)^{2}}}}$ 

This formula only works for | r | < 1 as well. From this, it follows that, for | r | < 1,

${\displaystyle \sum _{k=0}^{\infty }kr^{k}={\frac {r}{\left(1-r\right)^{2}}}\,;\,\sum _{k=0}^{\infty }k^{2}r^{k}={\frac {r\left(1+r\right)}{\left(1-r\right)^{3}}}\,;\,\sum _{k=0}^{\infty }k^{3}r^{k}={\frac {r\left(1+4r+r^{2}\right)}{\left(1-r\right)^{4}}}}$ 

Complex numbers

The summation formula for geometric series remains valid even when the common ratio is a complex number. This fact can be used, along with Euler's formula, to calculate some sums of non-obvious geometric series, such as:

${\displaystyle \sum _{k=0}^{\infty }{\frac {\sin(kx)}{r^{k}}}={\frac {1}{2i}}\left[\sum _{k=0}^{\infty }\left({\frac {e^{ix}}{r}}\right)^{k}-\sum _{k=0}^{\infty }\left({\frac {e^{-ix}}{r}}\right)^{k}\right]}$ .

It is clear that this is just the difference of two geometric series. From here, it is straighforward formula application to calculate that

${\displaystyle \sum _{k=0}^{\infty }{\frac {\sin(kx)}{r^{k}}}={\frac {r\sin(x)}{1+r^{2}-3r\cos(x)}}}$ 

See also

• Infinite series
• Thomas Robert Malthus
• Arithmetic progression
• Harmonic series