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Sequences and series formulas1/20/2024 ![]() ![]() (3) Magic Square series: In recreational mathematics, a magic square of order ‘n’ is an arrangement of n 2 numbers, usually distinct integers, in a square, such that the n numbers in all rows, all columns, and both diagonals sum to the same constant. ![]() Using binomial coefficients, the formula can be expressed as The maximum number p of pieces that can be created with a given number of cuts n, where n ≥ 0, is given by the formula For example, three cuts across a pancake will produce six pieces if the cuts all meet at a common point, but seven if they do not. (e) The Lazy Caterer’s Sequence: Formally also known as the central polygonal numbers, it describes the maximum number of pieces (or bounded/unbounded regions) of a circle (a pancake or pizza is usually used to describe the situation) that can be made with a given number of straight cuts. (d) Hexagonal Numbers: Similarly, pictorially, the hexagonal numbers can be represented as below: Pictorially, the Pentangular numbers can be can be represented as below: A pentagonal is given by the formula:įor n ≥ 1. (c) Pentagonal number Series: A pentagonal number is a figurate number that extends the concept of triangular and square numbers to the pentagon. Pictorially, the triangular numbers can be represented as below: The nth triangle number is the number of dots or balls in a triangle with n dots on a side it is the sum of the n natural numbers from 1 to n. ![]() (b) Triangular number Series: A triangular number or triangle number counts the objects that can form an equilateral triangle. Pictorially, the square numbers can be represented as below: (a) Square Numbers Series: it is quite self explanatory: 1, 4,9,16,25,36,49… (2) Figurate Numbers series like square, triangular, pentagonal, hexagonal no. There are many counting problems in combinatorics whose solution is given by the Fibonacci Numbers. The sum of the first n Fibonacci numbers is equal to the Fibonacci number two further along the sequence minus 1.Mathematically, F 1 + F 2+F 3…….+F n = F n+2 -1. The Fibonacci numbers in the composite-number (i.e., non-prime) positions are also composite numbers. Two consecutive Fibonacci numbers do not have any common factor, which means that they are Co-prime or relatively prime to each other. The sum of any ten consecutive Fibonacci numbers is divisible by 11. There are many properties of Fibonacci series, only a few are listed below: When we take much larger pairs of consecutive Fibonacci numbers, their quotients get us ever closer to the actual value of the golden ratio. These increasingly larger quotients seem to surround, the actual value of the golden ratio. Now, consider the quotient of the somewhat larger pair of consecutive For example, the quotient of the relatively small pair of consecutive Fibonacci numbers: The larger the Fibonacci numbers, the closer their ratio of last two terms approaches the golden ratio. There is practically no end to where these numbers appear or be sighted.įibonacci numbers are very much connected to the famous ‘Golden Ratio’ or ‘Divine ratio’ whose value is equal to 1.618… The Fibonacci numbers can be found in connection with the arrangement of branches on various species of trees, as well as in the number of ancestors at every generation of the male bee on its family tree. The appearances in nature seem boundless. More stunningly, they appear in nature abundantly for example, the number of spirals of bracts on a pinecone is always a Fibonacci number, and, similarly, the number of spirals of bracts on a pineapple is also a Fibonacci number. They pop up every now and then in nature, geometry, algebra, number theory, Permutations and combinations and many other branches of mathematics. Yet there are no numbers in all of mathematics as all-pervading as the fabulous Fibonacci numbers. However a quick inspection shows that it begins with two1 s and continues to get succeeding terms by adding, each time, the last two numbers to get the next number (i.e., 1 + 1 = 2, 1 + 2 = 3, 2 + 3 = 5, and so on).īy itself, this is not very remarkable. (1) Fibonacci Series: Probably the most famous of all Mathematical sequences it goes like this- 1,1,2,3,5,8,13,21,34,55,89…Īt first glance one may wonder what makes this sequence of numbers so sacrosanct or important or famous. Only a few of the more famous mathematical sequences are mentioned here: The sequences are also found in many fields like Physics, Chemistry and Computer Science apart from different branches of Mathematics. Such sequences are a great way of mathematical recreation. The world of mathematical sequences and series is quite fascinating and absorbing. ![]()
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