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The arithmetic mean, or less precisely the average, of a list of n numbers x 1, x 2, . . . , x n is the sum of the numbers divided by n: + + +. The geometric mean is similar, except that it is only defined for a list of nonnegative real numbers, and uses multiplication and a root in place of addition and division:
An arithmetico-geometric series is a series that has terms which are each the product of an element of an arithmetic progression with the corresponding element of a geometric progression. Example: 3 + 5 2 + 7 4 + 9 8 + 11 16 + ⋯ = ∑ n = 0 ∞ ( 3 + 2 n ) 2 n . {\displaystyle 3+{5 \over 2}+{7 \over 4}+{9 \over 8}+{11 \over 16}+\cdots =\sum ...
In mathematics, a generating function is a representation of an infinite sequence of numbers as the coefficients of a formal power series.Generating functions are often expressed in closed form (rather than as a series), by some expression involving operations on the formal series.
Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical concept after basic arithmetic and geometry. It is in Babylonian mathematics that elementary arithmetic (addition, subtraction, multiplication, and division) first
Traditionally, the sequence of natural numbers started with 1 (0 was not even considered a number for the Ancient Greeks.) However, in the 19th century, set theorists and other mathematicians started including 0 ( cardinality of the empty set , i.e. 0 elements, where 0 is thus the smallest cardinal number ) in the set of natural numbers.
Roth's theorem on arithmetic progressions (infinite version): A subset of the natural numbers with positive upper density contains a 3-term arithmetic progression. An alternate, more qualitative, formulation of the theorem is concerned with the maximum size of a Salem–Spencer set which is a subset of [ N ] = { 1 , … , N } {\displaystyle [N ...
Arithmetic is an elementary branch of mathematics that studies numerical operations like addition, subtraction, multiplication, and division. In a wider sense, it also includes exponentiation, extraction of roots, and taking logarithms. Arithmetic systems can be distinguished based on the type of numbers they operate on.
The axiomatization of arithmetic provided by Peano axioms is commonly called Peano arithmetic. The importance of formalizing arithmetic was not well appreciated until the work of Hermann Grassmann , who showed in the 1860s that many facts in arithmetic could be derived from more basic facts about the successor operation and induction .
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