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
After looking at the graph I have come to the conclusion that as you keep continuing terms with the sequence the deference between each term begins to decrease. After looking at the graph I began to notice that after the 4th term the graph stopped rising and become constant with the an values. So you can see that the graph is slowly approach a horizontal asymptote. Also after considering the value of an – an+1 I also came to the conclusion that the difference between each keeps decreasing and if I were to look at terms greater than 10, the difference between each term would eventually become non-existent.
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Therefore, I have come to the conclusion that the exact value for this infinite surd must be around 1. 618 The exact value of the infinite surd can be proven by using the quadratic formula by making the infinite surd equal x. X=1+1+1 …. x2=1+1+12 x2=1+x x2-x-1=0 Now that I have come up with a quadratic equation I can use the quadratic formula to find the value of x which will equal also equal the exact value of the infinite surd. 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. =-b±b2-4ac2a