The exact value of the infinite surd is.
Only the positive is used as you cannot have a real negative surd. In addition, 1.61803398 which is very close to the value for however the gaps between the values for become smaller and the difference between and is approximately -3.72528E-06 which is a minute change compared to the gap between and the value at which . These results support the answer which was provided algebraically due to the small margin between the calculated and algebraic results.
Part Two:
Consider another infinite surd:
Where the first term is:
Repeat the entire process above to find the exact value for this surd.
A recursive rule is devised from the observations made between terms in the series.
The major observation which identified the difference and determined the change between values of was that the original surd is added to a 2 and then square rooted much like the first infinite surd which was provided.
A table of values is composed for values of n in addition to a graph to identify possible asymptotes.
It can be observed that the values seem to approach 2 as becomes larger. The graph has a very similar shape to the graph in Part One but with a sharper turn and the values for also have a similar pattern in that they gradually decrease and become very small as becomes larger. However, the values are different in that they become more distinct. E.g. the big gaps are bigger, smaller gaps are smaller. To solve, a similar method to the method used in Part one will be used.
The exact value for the infinite surd is 2 where. These results correspond to the calculated results validating the observation that the infinite surd approached 2 with greater values of.
Part Three: The General Infinite Surd -
Consider the General Infinite Surd where the first term is
Find an expression for the exact value of this general infinite surd in terms of.
However, only the positive is used as negative surds are not real and cannot be put in the place of.
The exact value for the General Infinite Surd can be expressed as where represents the value of the infinite surd.
The value of an infinite surd is not always an integer.
Find some values of that make the expression an integer. Find the general statement that represents all values for is for which the expression is an integer.
The first value of which is an integer is 2 which is equal to 2 when due to the results of Part Two. The next integer value would be equal to 3. It can be worked out using the value for the General Infinite Surd.
Find when is equal to 3.
The value for where the value of the infinite surd is equal to 1 can also be worked out.
Finally, the value for where the value of the infinite surd is equal to 4 can be worked out in order to see if there is a clear pattern.
The results are tabulated to see if there is a correlation between the value for and the integer value of the surd.
The table shows how there is a growth in the value of which correlates directly to the integer values of the infinite surd. The correlation can be represented by the formula or or with the General expression for Infinite surds.
When put into an excel spread sheet, and tested, the values for all returned with the matching values for. Those values of when substituted into the table of values for showed values which all approached integers which were the original values. Examples of these results are shown below.
Infinite Surds and their traits are all closely interconnected with each other. The value of the infinite surd and the value for in the infinite surd have a unique correlation which means that they can easily be calculated at any time as long as one variable is at hand. In addition, the general expression of an Infinite Surd can be used to calculate the value of any surd effortlessly.