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 = 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 = 1.618016542
a5 = 1+a4 a10 = 1+a9 = 1.616121206 = 1.618028597
Looking at the results of the graph I decided to consider the value of an – an+1 as n gets very large, an being a term in the sequence and an+1 is the term before an. a2 – a1 a7 – a6 1.553773974  1.414213562 1.617851291  1.617442799 = 0.139560412 = 0.000408492
a3 – a2 a8 – a7
1.598053182  1.553773974 1.617977531  1.617851291 = 0.044279208 = 0.00012624
a4 – a3 a9 – a8
1.611847754  1.598053182 1.618016542  1.617977531
= 0.013794572 = 0.000039011
a5 – a4 a10 – a9
1.616121206  1.611847754 1.618028597  1.618016542
= 0.004273452 = 0.000012055
a6 – a5
1.617442799 – 1.616121206 = 0.001321593
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 nonexistent. 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...