Search results
Results from the WOW.Com Content Network
Moreover, every positive integer can be written in a unique way as the sum of one or more distinct Fibonacci numbers in such a way that the sum does not include any two consecutive Fibonacci numbers. This is known as Zeckendorf's theorem , and a sum of Fibonacci numbers that satisfies these conditions is called a Zeckendorf representation.
This list of mathematical series contains formulae for finite and infinite sums. It can be used in conjunction with other tools for evaluating sums. Here, is taken to have the value
(the Fibonacci sequence) is formed by starting with 0 and 1 and then adding any two consecutive terms to obtain the next one: an implicit description (sequence A000045 in the OEIS). The sequence 0, 3, 8, 15, ... is formed according to the formula n 2 − 1 for the n th term: an explicit definition.
The summation of an explicit sequence is denoted as a succession of additions. For example, summation of [1, 2, 4, 2] is denoted 1 + 2 + 4 + 2, and results in 9, that is, 1 + 2 + 4 + 2 = 9. Because addition is associative and commutative, there is no need for parentheses, and the result is the same irrespective of the order of the summands ...
Book IX, Proposition 36 of Elements proves that if the sum of the first n terms of this progression is a prime number (and thus is a Mersenne prime as mentioned above), then this sum times the n th term is a perfect number. For example, the sum of the first 5 terms of the series 1 + 2 + 4 + 8 + 16 = 31, which is a prime number. The sum 31 ...
The product of the members of a finite arithmetic progression with an initial element a 1, common differences d, and n elements in total is determined in a closed expression a 1 a 2 a 3 ⋯ a n = a 1 ( a 1 + d ) ( a 1 + 2 d ) . . .
This expression says that the output function f will be 1 for the minterms ,,,, and (denoted by the 'm' term) and that we don't care about the output for and combinations (denoted by the 'd' term). The summation symbol ∑ {\displaystyle \sum } denotes the logical sum (logical OR, or disjunction) of all the terms being summed over.
Diagram illustrating three basic geometric sequences of the pattern 1(r n−1) up to 6 iterations deep.The first block is a unit block and the dashed line represents the infinite sum of the sequence, a number that it will forever approach but never touch: 2, 3/2, and 4/3 respectively.