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Triangular Pyramidal Number
A tetrahedral number, or triangular pyramidal number, is a figurate number that represents a pyramid with a triangular base and three sides, called a tetrahedron. The th tetrahedral number, , is the sum of the first triangular numbers, that is, : Te_n = \sum_^n T_k = \sum_^n \frac = \sum_^n \left(\sum_^k i\right) The tetrahedral numbers are: : 1, 4, 10, 20, 35, 56, 84, 120, 165, 220, ... Formula The formula for the th tetrahedral number is represented by the 3rd rising factorial of divided by the factorial of 3: :Te_n= \sum_^n T_k = \sum_^n \frac = \sum_^n \left(\sum_^k i\right)=\frac = \frac The tetrahedral numbers can also be represented as binomial coefficients: :Te_n=\binom. Tetrahedral numbers can therefore be found in the fourth position either from left or right in Pascal's triangle. Proofs of formula This proof uses the fact that the th triangular number is given by :T_n=\frac. It proceeds by induction. ;Base case :Te_1 = 1 = \frac. ;Inductive step :\b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pyramid Of 35 Spheres Animation
A pyramid (from el, πυραμίς ') is a structure whose outer surfaces are triangular and converge to a single step at the top, making the shape roughly a pyramid in the geometric sense. The base of a pyramid can be trilateral, quadrilateral, or of any polygon shape. As such, a pyramid has at least three outer triangular surfaces (at least four faces including the base). The square pyramid, with a square base and four triangular outer surfaces, is a common version. A pyramid's design, with the majority of the weight closer to the ground and with the pyramidion at the apex, means that less material higher up on the pyramid will be pushing down from above. This distribution of weight allowed early civilizations to create stable monumental structures. Civilizations in many parts of the world have built pyramids. The largest pyramid by volume is the Great Pyramid of Cholula, in the Mexican state of Puebla. For thousands of years, the largest structures on Earth were pyramid ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gosper's Algorithm
In mathematics, Gosper's algorithm, due to Bill Gosper, is a procedure for finding sums of hypergeometric terms that are themselves hypergeometric terms. That is: suppose one has ''a''(1) + ... + ''a''(''n'') = ''S''(''n'') − ''S''(0), where ''S''(''n'') is a hypergeometric term (i.e., ''S''(''n'' + 1)/''S''(''n'') is a rational function of ''n''); then necessarily ''a''(''n'') is itself a hypergeometric term, and given the formula for ''a''(''n'') Gosper's algorithm finds that for ''S''(''n''). Outline of the algorithm Step 1: Find a polynomial ''p'' such that, writing ''b''(''n'') = ''a''(''n'')/''p''(''n''), the ratio ''b''(''n'')/''b''(''n'' − 1) has the form ''q''(''n'')/''r''(''n'') where ''q'' and ''r'' are polynomials and no ''q''(''n'') has a nontrivial factor with ''r''(''n'' + ''j'') for ''j'' = 0, 1, 2, ... . (This is always possible, whether or not the series is summable in cl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
The Twelve Days Of Christmas Visualisation
''The'' () is a grammatical article in English, denoting persons or things already mentioned, under discussion, implied or otherwise presumed familiar to listeners, readers, or speakers. It is the definite article in English. ''The'' is the most frequently used word in the English language; studies and analyses of texts have found it to account for seven percent of all printed English-language words. It is derived from gendered articles in Old English which combined in Middle English and now has a single form used with pronouns of any gender. The word can be used with both singular and plural nouns, and with a noun that starts with any letter. This is different from many other languages, which have different forms of the definite article for different genders or numbers. Pronunciation In most dialects, "the" is pronounced as (with the voiced dental fricative followed by a schwa) when followed by a consonant sound, and as (homophone of pronoun ''thee'') when followed by a v ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parity (mathematics)
In mathematics, parity is the property of an integer of whether it is even or odd. An integer is even if it is a multiple of two, and odd if it is not.. For example, −4, 0, 82 are even because \begin -2 \cdot 2 &= -4 \\ 0 \cdot 2 &= 0 \\ 41 \cdot 2 &= 82 \end By contrast, −3, 5, 7, 21 are odd numbers. The above definition of parity applies only to integer numbers, hence it cannot be applied to numbers like 1/2 or 4.201. See the section "Higher mathematics" below for some extensions of the notion of parity to a larger class of "numbers" or in other more general settings. Even and odd numbers have opposite parities, e.g., 22 (even number) and 13 (odd number) have opposite parities. In particular, the parity of zero is even. Any two consecutive integers have opposite parity. A number (i.e., integer) expressed in the decimal numeral system is even or odd according to whether its last digit is even or odd. That is, if the last digit is 1, 3, 5, 7, or 9, then it is odd; otherwis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Telescoping Series
In mathematics, a telescoping series is a series whose general term t_n can be written as t_n=a_n-a_, i.e. the difference of two consecutive terms of a sequence (a_n). As a consequence the partial sums only consists of two terms of (a_n) after cancellation. The cancellation technique, with part of each term cancelling with part of the next term, is known as the method of differences. For example, the series :\sum_^\infty\frac (the series of reciprocals of pronic numbers) simplifies as :\begin \sum_^\infty \frac & = \sum_^\infty \left( \frac - \frac \right) \\ & = \lim_ \sum_^N \left( \frac - \frac \right) \\ & = \lim_ \left\lbrack \right\rbrack \\ & = \lim_ \left\lbrack \right\rbrack \\ & = \lim_ \left\lbrack \right\rbrack = 1. \end An early statement of the formula for the sum or partial sums of a telescoping series can be found in a 1644 work by Evangelista Torricelli, ''De dimensione parabolae''. In general Telescoping sums are finite sums in which pairs ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Infinite Sum
In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, mathematical analysis. Series are used in most areas of mathematics, even for studying finite structures (such as in combinatorics) through generating functions. In addition to their ubiquity in mathematics, infinite series are also widely used in other quantitative disciplines such as physics, computer science, statistics and finance. For a long time, the idea that such a potentially infinite summation could produce a finite result was considered paradoxical. This paradox was resolved using the concept of a limit during the 17th century. Zeno's paradox of Achilles and the tortoise illustrates this counterintuitive property of infinite sums: Achilles runs after a tortoise, but when he reaches the position of the tortoise at the beginning of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Perfect Cube
In arithmetic and algebra, the cube of a number is its third power, that is, the result of multiplying three instances of together. The cube of a number or any other mathematical expression is denoted by a superscript 3, for example or . The cube is also the number multiplied by its square: :. The ''cube function'' is the function (often denoted ) that maps a number to its cube. It is an odd function, as :. The volume of a geometric cube is the cube of its side length, giving rise to the name. The inverse operation that consists of finding a number whose cube is is called extracting the cube root of . It determines the side of the cube of a given volume. It is also raised to the one-third power. The graph of the cube function is known as the cubic parabola. Because the cube function is an odd function, this curve has a center of symmetry at the origin, but no axis of symmetry. In integers A cube number, or a perfect cube, or sometimes just a cube, is a number whic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pollock Tetrahedral Numbers Conjecture
Pollock's conjectures are two closely related unproven conjectures in additive number theory. They were first stated in 1850 by Sir Frederick Pollock, better known as a lawyer and politician, but also a contributor of papers on mathematics to the Royal Society. These conjectures are a partial extension of the Fermat polygonal number theorem to three-dimensional figurate numbers, also called polyhedral numbers. *Pollock tetrahedral numbers conjecture: Every positive integer is the sum of at most five tetrahedral numbers. The numbers that are not the sum of at most 4 tetrahedral numbers are given by the sequence 17, 27, 33, 52, 73, ..., of 241 terms, with 343867 being almost certainly the last such number. *Pollock octahedral numbers conjecture: Every positive integer is the sum of at most seven octahedral numbers. This conjecture has been proven for all but finitely many positive integers. *Polyhedral numbers conjecture: Let ''m'' be the number of vertices of a platonic solid “ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sir Frederick Pollock, 1st Baronet
Sir Jonathan Frederick Pollock, 1st Baronet, PC (23 September 1783 – 28 August 1870) was a British lawyer and Tory politician. Background and education Pollock was the son of saddler David Pollock, of Charing Cross, London, and the elder brother of Field Marshal Sir George Pollock, 1st Baronet. An elder brother, Sir David Pollock, was a judge in India. The Pollock family were a branch of that family of Balgray, Dumfriesshire; David Pollock's father was a burgess of Berwick-upon-Tweed, and his grandfather a yeoman of Durham. His business as a saddler was given the official custom of the royal family. Sir John Pollock, 4th Baronet, great-great-grandson of David Pollock, stated in Time's Chariot (1950) that David was, 'perhaps without knowing it', Pollock of Balgray, the senior line of the family (Pollock of Pollock or Pollock of that ilk) having died out. Pollock was educated at St Paul's School and Trinity College, Cambridge. He was Senior Wrangler at Cambridge University. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square Number
In mathematics, a square number or perfect square is an integer that is the square (algebra), square of an integer; in other words, it is the multiplication, product of some integer with itself. For example, 9 is a square number, since it equals and can be written as . The usual notation for the square of a number is not the product , but the equivalent exponentiation , usually pronounced as " squared". The name ''square'' number comes from the name of the shape. The unit of area is defined as the area of a unit square (). Hence, a square with side length has area . If a square number is represented by ''n'' points, the points can be arranged in rows as a square each side of which has the same number of points as the square root of ''n''; thus, square numbers are a type of figurate numbers (other examples being Cube (algebra), cube numbers and triangular numbers). Square numbers are non-negative. A non-negative integer is a square number when its square root is again an intege ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square Pyramidal Number
In mathematics, a pyramid number, or square pyramidal number, is a natural number that counts the number of stacked spheres in a pyramid with a square base. The study of these numbers goes back to Archimedes and Fibonacci. They are part of a broader topic of figurate numbers representing the numbers of points forming regular patterns within different shapes. As well as counting spheres in a pyramid, these numbers can be described algebraically as a sum of the first n positive square numbers, or as the values of a cubic polynomial. They can be used to solve several other counting problems, including counting squares in a square grid and counting acute triangles formed from the vertices of an odd regular polygon. They equal the sums of consecutive tetrahedral numbers, and are one-fourth of a larger tetrahedral number. The sum of two consecutive square pyramidal numbers is an octahedral number. History The pyramidal numbers were one of the few types of three-dimensional figurate num ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cardano's Formula
In algebra, a cubic equation in one variable is an equation of the form :ax^3+bx^2+cx+d=0 in which is nonzero. The solutions of this equation are called roots of the cubic function defined by the left-hand side of the equation. If all of the coefficients , , , and of the cubic equation are real numbers, then it has at least one real root (this is true for all odd-degree polynomial functions). All of the roots of the cubic equation can be found by the following means: * algebraically, that is, they can be expressed by a cubic formula involving the four coefficients, the four basic arithmetic operations and th roots (radicals). (This is also true of quadratic (second-degree) and quartic (fourth-degree) equations, but not of higher-degree equations, by the Abel–Ruffini theorem.) * trigonometrically * numerical approximations of the roots can be found using root-finding algorithms such as Newton's method. The coefficients do not need to be real numbers. Much of what is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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