Infinite Surds

| INFINITE SURDS | Ria Garg |

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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 = 1.553773974 a2 = 1+1.414213562 = 1.553773974 a3 = 1+1+1+1 = 1.598053182 a3 = 1+1.553773974 = 1.598053182

Looking at the pattern it is clear that there is an intergradation of the previous an value into the next an value in the sequence. Therefore I came to the realization that the formula for an+1 in terms of an must be…

an+1 = 1+an

Now that I have discovered a formula for the infinite surd I will calculate the decimal values of the first 10 terms of this sequence and then plot the values on a graph, in order to understand the relationship between the n and an

a1 = 1+1 a6 = 1+a5 = 1.414213562 = 1.617442799

a2 = 1+a1 a7 = 1+a6 = 1.553773974 = 1.617851291

a3 = 1+a2 a8 = 1+a7 = 1.598053182 = 1.617977531

a4 = 1+a3 a9 = 1+aδ = 1.611847754