Infinite Surds

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…
x=2+22…. x2=2+222 x2=2+x x2x2 =