Geometric progression

Diagram showing the geometric series 1 + 1/2 + 1/4 + 1/8 + . . . which converges to 2.

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. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, a sequence is an ordered list of objects (or events A number is an Abstract object, tokens of which are Symbols used in Counting and measuring. 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.

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. A scale factor is a number which scales, or multiplies some quantity

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

$a_n = r\,a_{n-1}$ for every integer $n\geq 1$

Generally, to check whether a given sequence is geometric, one simply checks whether successive entries in the sequence all have the same ratio. "Difference equation" redirects here It should not be confused with a Differential equation.

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. Exponential growth (including Exponential decay) occurs when the growth rate of a mathematical function is proportional to the function's current value Infinity (symbolically represented with ∞) comes from the Latin infinitas or "unboundedness
1, the progression is a constant sequence.
Between -1 and 1 but not zero, there will be exponential decay towards zero. A quantity is said to be subject to exponential decay if it decreases at a rate proportional to its value
−1, the progression is an alternating sequence (see alternating series)
Less than −1, there will be exponential growth towards infinity (positive and negative). In Mathematics, an alternating series is an Infinite series of the form \sum_{n=0}^\infty (-1^n\a_n with an Exponential growth (including Exponential decay) occurs when the growth rate of a mathematical function is proportional to the function's current value Infinity (symbolically represented with ∞) comes from the Latin infinitas or "unboundedness

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, . Exponential growth (including Exponential decay) occurs when the growth rate of a mathematical function is proportional to the function's current value A quantity is said to be subject to exponential decay if it decreases at a rate proportional to its value The word linear comes from the Latin word linearis, which means created by lines. In Mathematics, an arithmetic progression or arithmetic sequence is a Sequence of Numbers such that the difference of any two successive members . . (with common difference 11). This result was taken by T.R. Malthus as the mathematical foundation of his Principle of Population. Thomas Robert Malthus FRS (13 February 1766 – 23 December 1834 was an English political economist and demographer who expressed views 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, the logarithm of a number to a given base is the power or Exponent to which the base must be raised in order to produce

Geometric series

Main article: Geometric series

A geometric series is the sum of the numbers in a geometric progression:

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

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. In Mathematics, a geometric series is a series with a constant ratio between successive terms. Rearranging (for $r\ne1$ ) 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

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

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. In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change In Mathematics, a geometric series is a series with a constant ratio between successive terms. In Mathematics, a series is often represented as the sum of a Sequence of terms That is a series is represented as a list of numbers with Such a series converges if and only if the absolute value of the common ratio is less than one ( | r | < 1 ). ↔ In Mathematics, the absolute value (or modulus) of a Real number is its numerical value without regard to its sign. Its value can then be computed from the finite sum formulae

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

Since:

$r^\infty = 0$ (when | r |<1).

Then:

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

For example, using numerical values

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

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. In Mathematics, especially Functional analysis, a Banach algebra, named after Stefan Banach, is an Associative algebra A over the In Mathematics, the p -adic number systems were first described by Kurt Hensel in 1897 As in the case for a finite sum, we can differentiate to calculate formulae for related sums. For example,

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

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

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

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 often represented as the sum of a Sequence of terms That is a series is represented as a list of numbers with In Mathematics, a series (or sometimes also an Integral) is said to converge absolutely if the sum (or integral of the Absolute value of the

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

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

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, a geometric series is a series with a constant ratio between successive terms. In Mathematics, an alternating series is an Infinite series of the form \sum_{n=0}^\infty (-1^n\a_n with an In Mathematics, a series (or sometimes also an Integral) is said to converge absolutely if the sum (or integral of the Absolute value of the

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

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

Complex numbers

The summation formula for geometric series remains valid even when the common ratio is a complex number. In Mathematics, a geometric series is a series with a constant ratio between successive terms. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted This fact can be used to calculate some sums of non-obvious geometric series, such as:

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

The proof of this formula starts with

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

a consequence of Euler's formula. This article is about Euler's formula in Complex analysis. For Euler's formula in algebraic topology and polyhedral combinatorics see Euler characteristic Substituting this into the series above, we get

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

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. The geometric mean in Mathematics, is a type of Mean or Average, which indicates the central tendency or typical value of a set of numbers (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. In Mathematics, an arithmetic progression or arithmetic sequence is a Sequence of Numbers such that the difference of any two successive members 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 )

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

$P=a^{n+1} r^{\frac{n(n+1)}{2}}$ . In Mathematics, an arithmetic progression or arithmetic sequence is a Sequence of Numbers such that the difference of any two successive members

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

We raise both sides to the second power:

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

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

Books VIII and IX of Euclid's Elements analyze geometric progressions and give several of their properties. Euclid ( Greek:.) fl 300 BC also known as Euclid of Alexandria, is often referred to as the Father of Geometry Euclid's Elements ( Greek:) is a mathematical and geometric Treatise consisting of 13 books written by the Greek

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 Quantity expressing the two- Dimensional size of a defined part of a Surface, typically a region bounded by a closed Curve. Geometry 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 any solid plasma vacuum or theoretical object is how much three- Dimensional 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, . The binary numeral system, or base-2 number system, is a Numeral system that represents numeric values using two symbols usually 0 and 1. . . ) is important in number theory. 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 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. In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1 In mathematics a perfect number is defined as a positive integer which is the sum of its proper positive Divisors that is the sum of the positive divisors excluding 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 in the series) equals 496, which is a perfect 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. In Mathematics, a divisor of an Integer n, also called a factor of n, is an integer which evenly divides n without 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. In Number theory, the fundamental theorem of arithmetic (or unique-prime-factorization theorem) states that every Natural number greater than 1 can be written 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.

References

Hall & Knight, Higher Algebra, p. In Mathematics, an arithmetic progression or arithmetic sequence is a Sequence of Numbers such that the difference of any two successive members The exponential function is a function in Mathematics. The application of this function to a value x is written as exp( x) See Harmonic series (music for the (related musical concept In Mathematics, the harmonic series is the Infinite series In Mathematics, a series is often represented as the sum of a Sequence of terms That is a series is represented as a list of numbers with Thomas Robert Malthus FRS (13 February 1766 – 23 December 1834 was an English political economist and demographer who expressed views Hackenbush is a two-player Mathematical game which may be played on any configuration of colored Line segments connected to one another by their endpoints and to the 39, ISBN 81-8116-000-2
Eric W. Weisstein, Geometric Series at MathWorld. Eric W Weisstein (born March 18, 1969, in Bloomington Indiana) is an Encyclopedist who created and maintains MathWorld MathWorld is an online Mathematics reference work created and largely written by Eric W

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