Infinite Surds

1853 words 8 pages


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 =
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Looking at the graph after term three the graph stops elevating and approaches a horizontal asymptote. Also after looking at the results of an – an+1 I have also come to the conclusion that as the term gets larger the difference between each term gets smaller and smaller. If I were to go past term 10 the difference between each term would eventually come to zero. Therefore, I believe that the exact value of this sequence of infinite surds must be 2 and I can prove this by using the quadratic formula.

x=2+22…. x2=2+222 x2=2+x x2-x-2 =