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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a geometric progression, also known as a geometric sequence, is a
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called ...
of non-zero
number A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual number ...
s where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the ''common ratio''. For example, the sequence 2, 6, 18, 54, ... is a geometric progression with common ratio 3. Similarly 10, 5, 2.5, 1.25, ... is a geometric sequence with common ratio 1/2. Examples of a geometric sequence are powers ''r''''k'' of a fixed non-zero number ''r'', such as 2''k'' and 3''k''. The general form of a geometric sequence is :a,\ ar,\ ar^2,\ ar^3,\ ar^4,\ \ldots where ''r'' ≠ 0 is the common ratio and ''a'' ≠ 0 is a scale factor, equal to the sequence's start value. The sum of a geometric progression terms is called a '' geometric series''.


Elementary properties

The ''n''-th term of a geometric sequence with initial value ''a'' = ''a''1 and common ratio ''r'' is given by :a_n = a\,r^, and in general :a_n = a_m\,r^. Such a geometric sequence also follows the
recursive relation In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a paramet ...
:a_n = r\,a_ for every integer 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 common ratio −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 Exponential growth is a process that increases quantity over time. It occurs when the instantaneous rate of change (that is, the derivative) of a quantity with respect to time is proportional to the quantity itself. Described as a function, a ...
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 A quantity is subject to exponential decay if it decreases at a rate proportional to its current value. Symbolically, this process can be expressed by the following differential equation, where is the quantity and (lambda) is a positive rate ...
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 common ratio not equal to −1, 1 or 0) show exponential growth or exponential decay, as opposed to the
linear Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...
growth (or decline) of an
arithmetic progression An arithmetic progression or arithmetic sequence () is a sequence of numbers such that the difference between the consecutive terms is constant. For instance, the sequence 5, 7, 9, 11, 13, 15, . . . is an arithmetic progression with a common differ ...
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 In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number  to the base  is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 ...
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: ::b^2=ac where ''b'' is considered to be the ''geometric mean'' between ''a'' and ''c''.


Geometric series


Product

The product of a geometric progression is the product of all terms. It can be quickly computed by taking the
geometric mean In mathematics, the geometric mean is a mean or average which indicates a central tendency of a set of numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum). The geometric mean is defined as the ...
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 In mathematics and statistics, the arithmetic mean ( ) or arithmetic average, or just the '' mean'' or the ''average'' (when the context is clear), is the sum of a collection of numbers divided by the count of numbers in the collection. The co ...
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: :\prod_^ ar^i = (\sqrt)^ = (\sqrt)^. (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 , it cannot produce a complex result if neither nor has an imaginary part. It is possible, should be negative and 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 \textstyle n + 1, which must be an even number because 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 represent the product. By definition, one calculates it by explicitly multiplying each individual term together. Written out in full, :P = a \cdot ar \cdot ar^2 \cdots ar^ \cdot ar^n. Carrying out the multiplications and gathering like terms, :P = a^ r^. The exponent of is the sum of an arithmetic sequence. Substituting the formula for that calculation, :P = a^ r^\frac, which enables simplifying the expression to :P = (ar^\frac)^ = (a\sqrt)^. Rewriting as \textstyle \sqrt, :P = (\sqrt)^, which concludes the proof.


History

A clay tablet from the Early Dynastic Period in Mesopotamia, MS 3047, contains a geometric progression with base 3 and multiplier 1/2. It has been suggested to be
Sumer Sumer () is the earliest known civilization in the historical region of southern Mesopotamia (south-central Iraq), emerging during the Chalcolithic and early Bronze Ages between the sixth and fifth millennium BC. It is one of the cradles of ...
ian, from the city of
Shuruppak Shuruppak ( sux, , "the healing place"), modern Tell Fara, was an ancient Sumerian city situated about 55 kilometres (35 mi) south of Nippur on the banks of the Euphrates in Iraq's Al-Qādisiyyah Governorate. Shuruppak was dedicated to ...
. It is the only known record of a geometric progression from before the time of Babylonian mathematics. Books VIII and IX of
Euclid Euclid (; grc-gre, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of ...
's ''Elements'' analyzes geometric progressions (such as the
powers of two A power of two is a number of the form where is an integer, that is, the result of exponentiation with number two as the base and integer  as the exponent. In a context where only integers are considered, is restricted to non-negati ...
, see the article for details) and give several of their properties.


See also

* * * * * * * * * *


References

*Hall & Knight, ''Higher Algebra'', p. 39,


External links

*
Derivation of formulas for sum of finite and infinite geometric progression
at Mathalino.com
Geometric Progression Calculator


a
sputsoft.com
* {{Series (mathematics) Sequences and series Mathematical series Articles containing proofs