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Tetrahedral Numbers
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 : ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pyramid Of 35 Spheres Animation
A pyramid () is a Nonbuilding structure, structure whose visible surfaces are triangular in broad outline and converge toward the top, making the appearance roughly a Pyramid (geometry), pyramid in the geometric sense. The base of a pyramid can be of any polygon shape, such as triangular or quadrilateral, and its surface-lines either filled or stepped. A pyramid has the majority of its mass closer to the ground with less mass towards the pyramidion at the Apex (geometry), apex. This is due to the gradual decrease in the cross-sectional area along the vertical axis with increasing elevation. This offers a weight distribution that allowed early civilizations to create monumental structures.Ancient Civilization, civilizations in many parts of the world pioneered the building of pyramids. The largest pyramid by volume is the Mesoamerican Great Pyramid of Cholula, in the Mexican state of Puebla. For millennia, the List of largest buildings in the world, largest structures on Earth ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gosper's Algorithm
In mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ..., Gosper's algorithm, due to Bill Gosper, is a procedure for finding sums of Hypergeometric identities, 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'' −&n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tetrahedral Triangular Number 10
In geometry, a tetrahedron (: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertices. The tetrahedron is the simplest of all the ordinary convex polyhedra. The tetrahedron is the three-dimensional case of the more general concept of a Euclidean simplex, and may thus also be called a 3-simplex. The tetrahedron is one kind of pyramid, which is a polyhedron with a flat polygon base and triangular faces connecting the base to a common point. In the case of a tetrahedron, the base is a triangle (any of the four faces can be considered the base), so a tetrahedron is also known as a "triangular pyramid". Like all convex polyhedra, a tetrahedron can be folded from a single sheet of paper. It has two such nets. For any tetrahedron there exists a sphere (called the circumsphere) on which all four vertices lie, and another sphere (the insphere) tangent to the tetrahedron's faces. Reg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parity (mathematics)
In mathematics, parity is the Property (mathematics), property of an integer of whether it is even or odd. An integer is even if it is divisible by 2, and odd if it is not.. For example, −4, 0, and 82 are even numbers, while −3, 5, 23, and 69 are odd numbers. The above definition of parity applies only to integer numbers, hence it cannot be applied to numbers with decimals or fractions like 1/2 or 4.6978. 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; otherwise it is even—as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Telescoping Series
In mathematics, a telescoping series is a series whose general term t_n is of the form t_n=a_-a_n, i.e. the difference of two consecutive terms of a sequence (a_n). As a consequence the partial sums of the series 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. 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''. Definition Telescoping sums are finite sums in which pairs of consecutive terms partly cancel each other, leaving only parts of the initial and final terms. Let a_n be the elements of a sequence of numbers. Then \sum_^N \left(a_n - a_\right) = a_N - a_0. If a_n converges to a limit L, the telescoping series gives: \sum_^\infty \left(a_n - a_\right) = L-a_0. Every series is a telescoping series of its own parti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Infinite Sum
In mathematics, a series is, roughly speaking, an addition of infinitely many terms, one after the other. 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 in combinatorics through generating functions. The mathematical properties of infinite series make them widely applicable in other quantitative disciplines such as physics, computer science, statistics and finance. Among the Ancient Greeks, the idea that a potentially infinite summation could produce a finite result was considered paradoxical, most famously in Zeno's paradoxes. Nonetheless, infinite series were applied practically by Ancient Greek mathematicians including Archimedes, for instance in the quadrature of the parabola. The mathematical side of Zeno's paradoxes was resolved using the concept of a limit during the 17th century, especially through the early calculus of Isaac Newton. 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 is denoted , using a superscript 3, for example . The cube operation can also be defined for any other mathematical expression, for example . 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pollock Tetrahedral Numbers Conjecture
Pollock's conjectures are closely related 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. Statement of the conjectures *Pollock tetrahedral numbers conjecture: Every positive integer is the sum of at most 5 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 343,867 conjectured to be the last such number. *Pollock octahedral numbers conjecture: Every positive integer is the sum of at most 7 octahedral numbers. This conjecture has been proven for all but finitely many positive integers. *Pollock cube numbers conjecture: Every positive integer is the sum of at most 9 cub ... [...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 to HM King George III 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 ... [...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 number, figurate numbers (other examples being Cube (algebra), cube numbers and triangular numbers). In the Real number, real number system, square numbers are non-negative. A non-negative integer ... [...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 stacked spheres in a pyramid (geometry), 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 fi ... [...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 not zero. The solutions of this equation are called root of a function, 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: * Algebra, algebraically: more precisely, they can be expressed by a ''cubic formula'' involving the four coefficients, the four basic arithmetic operations, square roots, and cube roots. (This is also true of quadratic equation, quadratic (second-degree) and quartic equation, quartic (fourth-degree) equations, but not for higher-degree equations, by the Abel–Ruffini theorem.) * Trigonometry, trigonometrically * numerical approximations of the roots can be found using root-finding algorithm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
