# Geometric progression

A geometric progression, also known as a geometric sequence, is a mathematical sequence of non-zero numbers where each term after the first is found by multiplying the previous one by a fixed number called the common ratio. For example, the sequence 2, 6, 18, 54, ... is a geometric progression with a common ratio of 3. Similarly 10, 5, 2.5, 1.25, ... is a geometric sequence with a common ratio of 1/2.

Examples of a geometric sequence are powers rk of a fixed non-zero number r, such as 2k and 3k. The general form of a geometric sequence is

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

where r is the common ratio and a ≠ 0 is a scale factor, equal to the sequence's start value. The sum of a geometric progression's terms is called a geometric series.

## Elementary properties

The n-th term of a geometric sequence with initial value a = a1 and common ratio r is given by

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

and in general

${\displaystyle a_{n}=a_{m}\,r^{n-m}.}$

Such a geometric sequence also follows the recursive relation

${\displaystyle a_{n}=r\,a_{n-1}}$ for every integer ${\displaystyle n\geq 2.}$

Generally, to check whether a given sequence is geometric, one simply checks whether successive entries in the sequence all have the same ratio.

The common ratio of a geometric sequence may be negative, resulting in an alternating sequence, with numbers alternating between positive and negative. For instance

1, −3, 9, −27, 81, −243, ...

is a geometric sequence with a common ratio of −3.

The behaviour of a geometric sequence 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.
• greater than 1, there will be exponential growth towards positive or negative infinity (depending on the sign of the initial term).
• 1, the progression is a constant sequence.
• between −1 and 1 but not zero, there will be exponential decay towards zero (→ 0).
• −1, the absolute value of each term in the sequence is constant and terms alternate in sign.
• less than −1, for the absolute values there is exponential growth towards (unsigned) infinity, due to the alternating sign.

Geometric sequences (with a common ratio not equal to −1, 1 or 0) show exponential growth or exponential decay, as opposed to the linear growth (or decline) of an arithmetic progression such as 4, 15, 26, 37, 48, … (with common difference 11). 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: exponentiating each term of an arithmetic progression yields a geometric progression while taking the logarithm of each term in a geometric progression with a positive common ratio yields an arithmetic progression.

An interesting result of the definition of the geometric progression is that any three consecutive terms a, b and c will satisfy the following equation:

${\displaystyle b^{2}=ac}$

where b is considered to be the geometric mean between a and c.

## Geometric series

In mathematics, a geometric series is a series in which the ratio of successive adjacent terms is constant. In other words, the sum of consecutive terms of a geometric sequence forms a geometric series. Each term is therefore the geometric mean of its two neighbouring terms, similar to how the terms in an arithmetic series are the arithmetic means of their two neighbouring terms.

Geometric series have been studied in mathematics from at least the time of Euclid in his work, Elements, which explored geometric proportions. Archimedes further advanced the study through his work on infinite sums, particularly in calculating areas and volumes of geometric shapes (for instance calculating the area inside a parabola) and the early development of calculus, where they have been paradigmatic examples of both convergent series and divergent series. They serve as prototypes for frequently used mathematical tools such as Taylor series, Fourier series, and matrix exponentials.

Geometric series have been applied to model a wide variety of natural phenomena and social phenomena, such as the expansion of the universe where the common ratio between terms is defined by Hubble's constant, the decay of radioactive carbon-14 atoms where the common ratio between terms is defined by the half-life of carbon-14, probabilities of winning in games of chance where the common ratio could be determined by the odds of a roulette wheel, and the economic values of investments where the common ratio could be determined by a combination of inflation rates and interest rates.

In general, a geometric series is written as ${\displaystyle a+ar+ar^{2}+ar^{3}+...}$, where ${\displaystyle a}$ is the initial term and ${\displaystyle r}$ is the common ratio between adjacent terms. For example, the series

${\displaystyle {\frac {1}{2}}\,+\,{\frac {1}{4}}\,+\,{\frac {1}{8}}\,+\,{\frac {1}{16}}\,+\,\cdots }$

is geometric because each successive term can be obtained by multiplying the previous term by ${\displaystyle 1/2}$.

Truncated geometric series ${\displaystyle a+ar+ar^{2}+ar^{3}+\dots +ar^{n}}$ are called "finite geometric series" in certain branches of mathematics, especially in 19th century calculus and in probability and statistics and their applications.

The standard generator form[1] expression for the infinite geometric series is

${\displaystyle a+ar+ar^{2}+ar^{3}+\dots =\sum _{k=0}^{\infty }ar^{k}}$

and the generator form expression for the finite geometric series is

${\displaystyle a+ar+ar^{2}+ar^{3}+\dots +ar^{n}=\sum _{k=0}^{n}ar^{k}.}$

Any finite geometric series has the sum ${\displaystyle a(1-r^{n+1})/(1-r)}$, and when ${\displaystyle |r|<1}$ the infinite series converges to the value ${\displaystyle a/(1-r)}$.

Though geometric series are most commonly found and applied with the real or complex numbers for ${\displaystyle a}$ and ${\displaystyle r}$, there are also important results and applications for matrix-valued geometric series, function-valued geometric series, p-adic number geometric series, and, most generally, geometric series of elements of abstract algebraic fields, rings, and semirings.

## Product

The product of a geometric progression is the product of all terms. It can be quickly computed by taking the geometric mean of the progression's first and last individual terms, and raising that mean to the power given by the number of terms. (This is very similar to the formula for the sum of terms of an arithmetic sequence: take the arithmetic mean of the first and last individual terms, and multiply by the number of terms.)

As the geometric mean of two numbers equals the square root of their product, the product of a geometric progression is:

${\displaystyle \prod _{i=0}^{n}ar^{i}=({\sqrt {a\cdot ar^{n}}})^{n+1}=({\sqrt {a^{2}r^{n}}})^{n+1}}$.

(An interesting aspect of this formula is that, even though it involves taking the square root of a potentially-odd power of a potentially negative r, it cannot produce a complex result if neither a nor r has an imaginary part. It is possible, should r be negative and n be odd, for the square root to be taken of a negative intermediate result, causing a subsequent intermediate result to be an imaginary number. However, an imaginary intermediate formed in that way will soon afterwards be raised to the power of ${\displaystyle \textstyle n+1}$, which must be an even number because n by itself was odd; thus, the final result of the calculation may plausibly be an odd number, but it could never be an imaginary one.)

### Proof

Let P represent the product. By definition, one calculates it by explicitly multiplying each individual term together. Written out in full,

${\displaystyle P=a\cdot ar\cdot ar^{2}\cdots ar^{n-1}\cdot ar^{n}}$.

Carrying out the multiplications and gathering like terms,

${\displaystyle P=a^{n+1}r^{1+2+3+\cdots +(n-1)+n}}$.

The exponent of r is the sum of an arithmetic sequence. Substituting the formula for that calculation,

${\displaystyle P=a^{n+1}r^{\frac {n(n+1)}{2}}}$,

which enables simplifying the expression to

${\displaystyle P=(ar^{\frac {n}{2}})^{n+1}=(a{\sqrt {r^{n}}})^{n+1}}$.

Rewriting a as ${\displaystyle \textstyle {\sqrt {a^{2}}}}$,

${\displaystyle P=({\sqrt {a^{2}r^{n}}})^{n+1}}$,

which concludes the proof.

## History

A clay tablet from the Early Dynastic Period in Mesopotamia (c. 2900 – c. 2350 BC), identified as MS 3047, contains a geometric progression with base 3 and multiplier 1/2. It has been suggested to be Sumerian, from the city of Shuruppak. It is the only known record of a geometric progression from before the time of old Babylonian mathematics beginning in 2000 BC.[2]

Books VIII and IX of Euclid's Elements analyze geometric progressions (such as the powers of two, see the article for details) and give several of their properties.[3]

## References

1. ^ Riddle, Douglas F. Calculus and Analytic Geometry, Second Edition Belmont, California, Wadsworth Publishing, p. 566, 1970.
2. ^ Friberg, Jöran (2007). "MS 3047: An Old Sumerian Metro-Mathematical Table Text". In Friberg, Jöran (ed.). A remarkable collection of Babylonian mathematical texts. Sources and Studies in the History of Mathematics and Physical Sciences. New York: Springer. pp. 150–153. doi:10.1007/978-0-387-48977-3. ISBN 978-0-387-34543-7. MR 2333050.
3. ^ Heath, Thomas L. (1956). The Thirteen Books of Euclid's Elements (2nd ed. [Facsimile. Original publication: Cambridge University Press, 1925] ed.). New York: Dover Publications.