NationMaster - Encyclopedia: Geometric progression

In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of numbers 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 and 10, 5, 2.5, 1.25, ... is a geometric sequence with common ratio 1/2. The sum of the terms of a geometric progression is known as a geometric series. Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... For other senses of this word, see sequence (disambiguation). ... For other uses, see Number (disambiguation). ... Addition is one of the basic operations of arithmetic. ...

Thus, the general form of a geometric sequence is

$a, ar, ar^2, ar^3, ar^4, ldots$

and that of a geometric series is

$a + ar + ar^2 + ar^3 + ar^4 + ldots$

where r ≠ 0 is the common ratio and a is a scale factor, equal to the sequence's start value. lol rofl taco hahaThere is also a nscale factor for the expansion of the Universe Scale factors are used in computer science when certain real world numbers need to be represented on a different scale in order to fit a required number format. ...

1 Elementary properties
2 Geometric series
- 2.1 Infinite geometric series
- 2.2 Complex numbers
3 Product
4 Relationship to geometry and Euclid's work
- 4.1 Elements, Book IX
5 See also
6 References

Elementary properties

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

$a_n = a,r^{n-1}$

Such a geometric sequence also follows the recursive relation In mathematics, a recurrence relation is an equation which defines a sequence recursively: each term of the sequence is defined as a function of the preceding terms. ...

a n = r a n - 1

for every integer $ngeq 2$

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 series may be negative, resulting in an alternating sequence, with numbers switching from positive to negative and back. 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 towards positive infinity.
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).

Geometric sequences (with 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. In mathematics, exponential growth (or geometric growth) occurs when the growth rate of a function is always proportional to the functions current size. ... The infinity symbol âˆž in several typefaces. ... A quantity is said to be subject to exponential decay if it decreases at a rate proportional to its value. ... In mathematics, an alternating series is an infinite series of the form with an â‰¥ 0. ... In mathematics, exponential growth (or geometric growth) occurs when the growth rate of a function is always proportional to the functions current size. ... The infinity symbol âˆž in several typefaces. ... In mathematics, exponential growth (or geometric growth) occurs when the growth rate of a function is always proportional to the functions current size. ... A quantity is said to be subject to exponential decay if it decreases at a rate proportional to its value. ... In mathematics, an arithmetic progression or arithmetic sequence is a sequence of numbers such that the difference of any two successive members of the sequence is a constant. ... The Rev. ... Logarithms to various bases: is to base e, is to base 10, and is to base 1. ...

Geometric series

Main article: Geometric series

A geometric series is the sum of the numbers in a geometric progression: In mathematics, a geometric progression 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. ...

$sum_{k=0}^{n} ar^k = ar^0+ar^1+ar^2+ar^3+cdots+ar^n ,$

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

$(1-r) sum_{k=0}^{n} ar^k = a-ar^{n+1},$

since all the other terms cancel. Rearranging (for $rne1$ ) gives the convenient formula for a geometric series:

$sum_{k=0}^{n} ar^k = frac{a(1-r^{n+1})}{1-r}$

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

$sum_{k=m}^n ar^k=frac{a(r^m-r^{n+1})}{1-r}$

Differentiating this formula with respect to r allows us to arrive at formulae for sums of the form For a non-technical overview of the subject, see Calculus. ...

$sum_{k=0}^n k^s r^k$

For example:

$frac{d}{dr}sum_{k=0}^nr^k = sum_{k=0}^nkr^{k-1}= frac{1-r^{n+1}}{(1-r)^2}-frac{(n+1)r^n}{1-r}$

For a geometric series containing only even powers of $r$ multiply by $(1 - r 2)$ :

$(1-r^2) sum_{k=0}^{n} ar^{2k} = a-ar^{2n+2}$

Then

$sum_{k=0}^{n} ar^{2k} = frac{a(1-r^{2n+2})}{1-r^2}$

For a series with only odd powers of $r$

$(1-r^2) sum_{k=0}^{n} ar^{2k+1} = ar-ar^{2n+3}$

and

$sum_{k=0}^{n} ar^{2k+1} = frac{ar(1-r^{2n+2})}{1-r^2}$

Infinite geometric series

Main article: 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 In mathematics, a geometric progression 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. ... In mathematics, a series is a sum of a sequence of terms. ... It has been suggested that this article or section be merged into Logical biconditional. ... In mathematics, the absolute value (or modulus[1]) of a real number is its numerical value without regard to its sign. ...

$sum_{k=0}^infty ar^k = lim_{ntoinfty}{sum_{k=0}^{n} ar^k} = lim_{ntoinfty}frac{a(1-r^{n+1})}{1-r} = lim_{ntoinfty}frac{a}{1-r} - lim_{ntoinfty}{frac{ar^{n+1}}{1-r}} = frac{a}{1-r}$

For a series containing only even powers of $r$ ,

$sum_{k=0}^infty ar^{2k} = frac{a}{1-r^2}$

and for odd powers only,

$sum_{k=0}^infty ar^{2k+1} = frac{ar}{1-r^2}$

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

$sum_{k=m}^infty ar^k=frac{ar^m}{1-r}$

Above 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, In functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers which at the same time is also a Banach space. ... The title given to this article is incorrect due to technical limitations. ...

$frac{d}{dr}sum_{k=0}^infty r^k = sum_{k=0}^infty kr^{k-1}= frac{1}{(1-r)^2}$

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

$sum_{k=0}^{infty} k r^k = frac{r}{left(1-rright)^2} ,;, sum_{k=0}^{infty} k^2 r^k = frac{r left( 1+r right)}{left(1-rright)^3} , ; , sum_{k=0}^{infty} k^3 r^k = frac{r left( 1+4 r + r^2right)}{left( 1-rright)^4}$

Also, the infinite series 1/2 + 1/4 + 1/8 + 1/16 + · · · is an elementary example of a series that converges absolutely. In mathematics, a series is a sum of a sequence of terms. ... In mathematics, a series is a sum of a sequence of terms. ...

It is a geometric series whose first term is 1/2 and whose common ratio is 1/2, so its sum is In mathematics, a geometric progression 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. ...

$frac12+frac14+frac18+frac{1}{16}+cdots=frac{1/2}{1-(+1/2)} = 1.$

The inverse of the above series is 1/2 − 1/4 + 1/8 − 1/16 + · · · is a simple example of an alternating series that converges absolutely. In mathematics, an alternating series is an infinite series of the form with an â‰¥ 0. ... In mathematics, a series is a sum of a sequence of terms. ...

It is a geometric series whose first term is 1/2 and whose common ratio is −1/2, so its sum is In mathematics, a geometric progression 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. ...

$frac12-frac14+frac18-frac{1}{16}+cdots=frac{1/2}{1-(-1/2)} = frac13.$

Complex numbers

The summation formula for geometric series remains valid even when the common ratio is a complex number. This fact can be used to calculate some sums of non-obvious geometric series, such as: In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = âˆ’1. ...

$sum_{k=0}^{infty} frac{sin(kx)}{r^k} = frac{r sin(x)}{1 + r^2 - 2 r cos(x)}$

The proof of this formula starts with

$sin(kx) = frac{e^{ikx} - e^{-ikx}}{2i}$

a consequence of Euler's formula. Substituting this into the series above, we get Eulers formula, named after Leonhard Euler, is a mathematical formula in complex analysis that shows a deep relationship between the trigonometric functions and the complex exponential function. ...

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

This is just the difference of two geometric series. From here, it is then a straightforward application of our formula for infinite geometric series to finish the proof.

Product

The product of a geometric progression is the product of all terms. If all terms are positive, then it can be quickly computed by taking the geometric mean of the progression's first and last term, 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 term and multiply with the number of terms.) The geometric mean of a collection of positive data is defined as the nth root of the product of all the members of the data set, where n is the number of members. ... This is a page about mathematics. ... In mathematics and statistics, the arithmetic mean (or simply the mean) of a list of numbers is the sum of all the members of the list divided by the number of items in the list. ...

$prod_{i=0}^{n} ar^i = left( sqrt{a_1 cdot a_{n+1}}right)^{n+1}$ (if

a, r > 0

Proof:

Let the product be represented by P:

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

Now, carrying out the multiplications, we conclude that

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

Applying the sum of arithmetic series, the expression will yield // Definition In mathematics, an arithmetic series is the sum of the components of an arithmetic progression. ...

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

$P=(ar^{frac{n}{2}})^{n+1}$ .

We raise both sides to the second power:

$P^2=(a^2 r^{n})^{n+1}=(acdot ar^n)^{n+1}$ .

Consequently

$P^2=(a_1 cdot a_{n+1})^{n+1}$ and

$P=(a_1 cdot a_{n+1})^{frac{n+1}{2}}$ ,

which concludes the proof.

Relationship to geometry and Euclid's work

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Books VIII and IX of Euclid's Elements analyze geometric progressions and give several of their properties. For other uses, see Euclid (disambiguation). ... The frontispiece of Sir Henry Billingsleys first English version of Euclids Elements, 1570 Euclids Elements (Greek: ) is a mathematical and geometric treatise, consisting of 13 books, written by the Hellenistic mathematician Euclid in Alexandria circa 300 BC. It comprises a collection of definitions, postulates (axioms), propositions (theorems...

A geometric progression gains its geometric character from the fact that the areas of two geometrically similar plane figures are in "duplicate" ratio to their corresponding sides; further the volumes of two similar solid figures are in "triplicate" ratio of their corresponding sides. Area is a physical quantity expressing the size of a part of a surface. ... // Two geometrical objects are called similar if one is congruent to the result of a uniform scaling (enlarging or shrinking) of the other. ... The volume of a solid object is the three-dimensional concept of how much space it occupies, often quantified numerically. ...

The meaning of the words "duplicate" and "triplicate" in the previous paragraph is illustrated by the following examples. Given two squares whose sides have the ratio 2 to 3, then their areas will have the ratio 4 to 9; we can write this as 4 to 6 to 9 and notice that the ratios 4 to 6 and 6 to 9 both equal 2 to 3; so by using the side ratio 2 to 3 "in duplicate" we obtain the ratio 4 to 9 of the areas, and the sequence 4, 6, 9 is a geometric sequence with common ratio 3/2. Similarly, give two cubes whose side ratio is 2 to 5, their volume ratio is 8 to 125, which can be obtained as 8 to 20 to 50 to 125, the original ratio 2 to 5 "in triplicate", yielding a geometric sequence with common ration 5/2.

Elements, Book IX

The geometric progression 1, 2, 4, 8, 16, 32, ... (or, in the binary numeral system, 1, 10, 100, 1000, 10000, 100000, ... ) is important in number theory. Book IX, Proposition 36 of Elements proves that if the sum of the first n terms of this progression is a prime number, then this sum times the nth term is a perfect number. For example, the sum of the first 5 terms of the series (1 + 2 + 4 + 8 + 16) is 31, which is a prime number. The sum 31 multiplied by 16 (the 5th term is the series) equals 496, which is a perfect number. The binary numeral system, or base-2 number system, is a numeral system that represents numeric values using two symbols, usually 0 and 1. ... Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study. ... In mathematics, a prime number (or a prime) is a natural number which has exactly two distinct natural number divisors: 1 and itself. ... In mathematics, a perfect number is defined as an integer which is the sum of its proper positive divisors, that is, the sum of the positive divisors not including the number. ...

Book IX, Proposition 35 proves that in a geometric series if the first term is subtracted from the second and last term in the sequence then as the excess of the second is to the first, so will the excess of the last be to all of those before it. (This is a restatement of our formula for geometric series from above.) Applying this to the geometric progression 31,62,124,248,496 (which results from 1,2,4,8,16 by multiplying all terms by 31), we see that 62 minus 31 is to 31 as 496 minus 31 is to the sum of 31,62,124,248. Therefore the numbers 1,2,4,8,16,31,62,124,248 add up to 496 and further these are all the numbers which divide 496. For suppose that P divides 496 and it is not amongst these numbers. Assume P×Q equals 16×31, or 31 is to Q as P is to 16. Now P cannot divide 16 or it would be amongst the numbers 1,2,4,8,16. Therefore 31 cannot divide Q. And since 31 does not divide Q and Q measures 496, the fundamental theorem of arithmetic implies that Q must divide 16 and be amongst the numbers 1,2,4,8,16. Let Q be 4, then P must be 124, which is impossible since by hypothesis P is not amongst the numbers 1,2,4,8,16,31,62,124,248. In mathematics, a divisor of an integer n, also called a factor of n, is an integer which evenly divides n without leaving a remainder. ... In number theory, the fundamental theorem of arithmetic (or unique factorization theorem) states that every natural number either is itself a prime number, or can be written as a unique product of prime numbers. ...

References

Hall & Knight, Higher Algebra, p. 39, ISBN 81-8116-000-2
Eric W. Weisstein, Geometric Series at MathWorld.

Categories: Articles needing additional references from May 2007 | Mathematical series | Calculus Dr. Eric W. Weisstein Encyclopedist Dr. Eric W. Weisstein (born March 18, 1969, in Bloomington, Indiana) is a noted encyclopedist in several technical areas of science and mathematics. ... MathWorld is an online mathematics reference work, sponsored by Wolfram Research Inc. ...

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