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In mathematics, summation is the addition of a sequence of numbers, called addends or summands; the result is their sum or total.Beside numbers, other types of values can be summed as well: functions, vectors, matrices, polynomials and, in general, elements of any type of mathematical objects on which an operation denoted "+" is defined.
The sum of the reciprocals of all the Fermat numbers (numbers of the form + ) (sequence A051158 in the OEIS) is irrational. The sum of the reciprocals of the pronic numbers (products of two consecutive integers) (excluding 0) is 1 (see Telescoping series).
is the sum of the n-th powers of all the roots, ... If f(n) is an arithmetic function (i.e. a complex-valued function of the integers or natural numbers), ...
Waring's problem asks whether for every natural number k there exists an associated positive integer s such that every natural number is the sum of at most sk th powers of natural numbers. The successive powers of the golden ratio φ obey the Fibonacci recurrence: φ n + 1 = φ n + φ n − 1 . {\displaystyle \varphi ^{n+1}=\varphi ^{n}+\varphi ...
A prime number is a natural number greater than 1 that is not a product of two smaller natural numbers. A000040: ... Sum of numbers in any row, column, ...
Commutativity: for all natural numbers a and b, a + b = b + a and a × b = b × a. [54] Existence of identity elements: for every natural number a, a + 0 = a and a × 1 = a. If the natural numbers are taken as "excluding 0", and "starting at 1", then for every natural number a, a × 1 = a. However, the "existence of additive identity element ...
We prove associativity by first fixing natural numbers a and b and applying induction on the natural number c. For the base case c = 0, (a + b) + 0 = a + b = a + (b + 0) Each equation follows by definition [A1]; the first with a + b, the second with b. Now, for the induction. We assume the induction hypothesis, namely we assume that for some ...
Triangular numbers are a type of figurate number, other examples being square numbers and cube numbers. The n th triangular number is the number of dots in the triangular arrangement with n dots on each side, and is equal to the sum of the n natural numbers from 1 to n. The sequence of triangular numbers, starting with the 0th triangular number, is