Infinite Surds. As we can see there are ten terms of this sequence where is the general term of the sequence when, is the first term of the sequence.
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Math Portfolio Infinite Surds Introduction a surd is an irrational number that can not be written as a fraction of two integers but can only be expressed using the root sign. Bellow an example of an infinite surd: This surd can be turned into a set of particular numbers sequence: 1.414213562 1.553773974 1.598053182 1.611847754 a51.616121207 1.617442799 1.617851291 1.617977531 1.618016542 As we can see there are ten terms of this sequence where is the general term of the sequence when, is the first term of the sequence...Etc. A formula has been defined for in terms of: (1) A graph has been plotted to show the relation between and. ...read more.
to this: x2=1+x The equation can be solved using the solution of a quadric equation: Whereas a=1, b=-1, c=-1 Two solutions for x were obtained: x=1.618033989 and x= -0.61803387 The negative value is ignored so x=1.618033989 which is the exact value for this infinite surd. Another condition of infinite surd can be taken into consideration to acknowledge the point more: Where the first term of the sequence is The first ten terms of the sequence are: 1.847759065 1.961570561 1.990369453 1.997590912 1.999397637 1.999849404 1.999962351 1.999990588 h10= 1.999999421 And the formula of the sequence is obtained according to bn+1 which is relative to the term bn: The graph bellow shows the relation It can be observed that when n gets bigger, bn attempt to reach the value 2 which is the exact value for this infinite surd. ...read more.
so: To find some values of k to make the expression an integer: We can see that 4k is an even number and 4k+1 is odd, so is an odd number if 4k+1 is a perfect square hence 1+ is an even number and possible to be divided by 2. As a result if 4k+1 is a perfect square we can obtain an integral number in the result. For example let k=2: k=3 not integer because13 is not a perfect square Thus we come to the conclusion that only limited values of k can be used to make the result an integer and those values are any value of k can make 4k+1 a perfect square such as k = 2,6,12...etc. ...read more.
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