Fibonacci numbers
... Problem: For all the given numbers x0 , x1 , . . . , xn≠1 , such that 1 ˛ xi ˛ m ˛ 1 000 000, check whether they may be presented as the sum of two Fibonacci numbers. Solution: Notice that only a few tens of Fibonacci numbers are smaller than the maximal m (exactly 31). We consider all the pairs. If ...
... Problem: For all the given numbers x0 , x1 , . . . , xn≠1 , such that 1 ˛ xi ˛ m ˛ 1 000 000, check whether they may be presented as the sum of two Fibonacci numbers. Solution: Notice that only a few tens of Fibonacci numbers are smaller than the maximal m (exactly 31). We consider all the pairs. If ...
... A tower is 50m high. Its shadow is x metres shorter when the sun’s altitude is 45o than when it was 300. Find the value of x. Q22. If the point (x , y) is equidistant from the points (a + b, b-a) and (a-b , a + b), then prove that b x = a y. Q23. Find the sum of the first 25 terms of an AP whose nth ...
Sums of Angles of Star Polygons and the Eulerian Numbers
... To end this paper, we present a possible application. We imagine that the data in the triangle of nk might be useful in some physics experiments. For example, suppose an electron is ejected into a black box through a window. Assume that there are a number of moving attractor-reflectors installed in ...
... To end this paper, we present a possible application. We imagine that the data in the triangle of nk might be useful in some physics experiments. For example, suppose an electron is ejected into a black box through a window. Assume that there are a number of moving attractor-reflectors installed in ...
Prime Numbers and How to Avoid Them
... The “Chinese Remainder Theorem” guarantees we can find infinitely many such numbers. The smallest is k= 201 44650 31451 65117 This is a “Sierpinski number”, which makes every term in the infinite sequence composite. Half are divisible by 3, a quarter divisible by 5, etc. ...
... The “Chinese Remainder Theorem” guarantees we can find infinitely many such numbers. The smallest is k= 201 44650 31451 65117 This is a “Sierpinski number”, which makes every term in the infinite sequence composite. Half are divisible by 3, a quarter divisible by 5, etc. ...
algo11
... When A wants to talk to B , how does B know that A is the real A, not an enemy imitating A Method I : a trivial method B may ask the name of A’s mother (a private ...
... When A wants to talk to B , how does B know that A is the real A, not an enemy imitating A Method I : a trivial method B may ask the name of A’s mother (a private ...
Rational and Irrational Numbers
... cannot be expressed as a fraction. Also, irrational numbers cannot be represented as terminating or repeating decimals. • Irrational numbers are non-terminating, nonrepeating decimals. ...
... cannot be expressed as a fraction. Also, irrational numbers cannot be represented as terminating or repeating decimals. • Irrational numbers are non-terminating, nonrepeating decimals. ...
A Nonlinear Expression for Fibonacci Numbers and Its Consequences
... First of all, it may be worth mentioning that a great many research works have been done for Fibonacci number sequence and the like during past 60 years. The vast literature may be found more easily in the Fibonacci Quarterly, the journal that started publication since 1963. Denote by N and Z the na ...
... First of all, it may be worth mentioning that a great many research works have been done for Fibonacci number sequence and the like during past 60 years. The vast literature may be found more easily in the Fibonacci Quarterly, the journal that started publication since 1963. Denote by N and Z the na ...
2003 Paper 3 Practice
... The temperature has risen by 13°C. The temperature has fallen by 24°C. The temperature has risen by 25°C. The temperature has risen by 24°C. ...
... The temperature has risen by 13°C. The temperature has fallen by 24°C. The temperature has risen by 25°C. The temperature has risen by 24°C. ...
Law of large numbers
In probability theory, the law of large numbers (LLN) is a theorem that describes the result of performing the same experiment a large number of times. According to the law, the average of the results obtained from a large number of trials should be close to the expected value, and will tend to become closer as more trials are performed.The LLN is important because it ""guarantees"" stable long-term results for the averages of some random events. For example, while a casino may lose money in a single spin of the roulette wheel, its earnings will tend towards a predictable percentage over a large number of spins. Any winning streak by a player will eventually be overcome by the parameters of the game. It is important to remember that the LLN only applies (as the name indicates) when a large number of observations are considered. There is no principle that a small number of observations will coincide with the expected value or that a streak of one value will immediately be ""balanced"" by the others (see the gambler's fallacy)