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The partial sums of the series 1 + 2 + 3 + 4 + 5 + 6 + ⋯ are 1, 3, 6, 10, 15, etc.The nth partial sum is given by a simple formula: = = (+). This equation was known ...
The idea becomes clearer by considering the general series 1 − 2x + 3x 2 − 4x 3 + 5x 4 − 6x 5 + &c. that arises while expanding the expression 1 ⁄ (1+x) 2, which this series is indeed equal to after we set x = 1.
Every prime number p divides a Fibonacci number that can be determined by the value of p modulo 5. If p is congruent to 1 or 4 modulo 5, then p divides F p−1, and if p is congruent to 2 or 3 modulo 5, then, p divides F p+1. The remaining case is that p = 5, and in this case p divides F p.
For instance, the sequence 5, 7, 9, 11, 13, 15, . . . is an arithmetic progression with a common difference of 2. If the initial term of an arithmetic progression is a 1 {\displaystyle a_{1}} and the common difference of successive members is d {\displaystyle d} , then the n {\displaystyle n} -th term of the sequence ( a n {\displaystyle a_{n ...
The 4–1–3–2 is a variation of the 4–1–2–1–2 and features a strong and talented defensive centre midfielder. This allows the remaining three midfielders to play further forward and more aggressively, and also allows them to pass back to their defensive mid when setting up a play or recovering from a counterattack.
[3] The divergence of the harmonic series was first proven in 1350 by Nicole Oresme. [2] [4] Oresme's work, and the contemporaneous work of Richard Swineshead on a different series, marked the first appearance of infinite series other than the geometric series in mathematics. [5] However, this achievement fell into obscurity. [6]
The most common derivative of 1,3,5-triazine is 1,3,5-triazine-2,4,6-triamine, commonly known as melamine or cyanuramide. Another important derivative is 1,3,5-triazine-2,4,6-triol better known as cyanuric acid. Cyanuric chloride (2,4,6-trichloro-1,3,5-triazine) is the starting point for the manufacture of many herbicides such as Simazine and ...
Name First elements Short description OEIS Mersenne prime exponents : 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, ... Primes p such that 2 p − 1 is prime.: A000043 ...