# Essay on Infinite Surds

1863 Words8 Pages
| INFINITE SURDS | Ria Garg | | The purpose of my investigation is to find the general statement that represents all values of k in an infinite surd for which the expression is an integer. I was able to achieve this goal through the process of going through various infinite surds and trying to find a relationship between each sequence. In the beginning stages of my investigation I came across the sequence of ` a1= 1+1 a2= 1+1+1 a3 = 1+1+1+1 While looking at the sequence I came to the realization that there is a very obvious pattern between each n value. The answer to each n value was plugged into the next n value. For example if you look at this sequence a1= 1+1 = 1.414213562 a2= 1+1+1 =…show more content…
Also since the sequence is a square root and, the graph shows no evidence of a root value I can disregard the negative answer to the infinite surd. x=-b±b2-4ac2a X=1+1-41-121 x=1+52 Now, I would like to carry on my investigation and look at another sequence of infinite surds where the first term of the sequence is 2+2. In the beginning steps of my investigation I came up with the formula an+1 = 1+an, which I will be using to further my investigation and find the first 10 terms of my new sequence. A1 = 2+2 a6 = 2+a5 = 1.847759065 = 1.999962351 a2 = 2+a1 a7 = 2+a6 = 1.961570561 = 1.999990588 a3 = 2+a2 a8 = 2+a7 = 1.990369453 = 1.999997647 a4 = 2+a3 a9 = 2+a8 = 1.997590912 = 1.999999412 a5 = 2+a4 a10 = 2+a9 = 1.999849404 = 1.999999853 Repeating the same process I completed with the previous sequence of infinite surds, for the next step I will consider the value of an – an+1