Infinite Surds Investigation. This graph illustrates the same relationship as was demonstrated in the infinite surd of 1. The largest jump between values occurs between 1 and 2.

       

The surd is canceled once the whole equation is squared and this is left:

Now this is solved using the quadratic formula:

The negative answer, , must be disregarded because a negative answer for a surd is not possible. Therefore the exact value of the infinite surd is:  

For the infinite surd expression:

This would be the sequence of terms for an:

                               etc.

Then the formula for an+1 in terms of an would be:

This formula is generated because it is known that an equals the term before an+1, and therefore by substituting in the previous value, an+1 can be determined:

,

Similarly                   therefore            

The decimal values of the first ten terms of the sequence are:

The graph for these values is:

It is interesting that the curves of both graphs look very similar. This graph illustrates the same relationship as was demonstrated in the infinite surd of 1. The largest jump between values occurs between 1 and 2. The gradient of the curve starts to approach zero as the value of n becomes larger as shown by value of an – an+1. This value can also be shown arithmetically:

 

The exact value of this infinite surd could be determined using the same idea as Part 1:

Substitute the value for an+1 into the equation and solve for an by setting it equal to zero.

       

The surd is canceled once the whole equation is squared and this is left:

This equation can be solved by factoring:

Again the negative answer, -1, must be disregarded because a surd cannot have a negative answer.

So the exact value of this infinite surd is: 2

For the infinite surd expression:

Where the first term is:

                       

The exact value of this general infinite surd can be found using the same idea as previously used:

Substitute the value for an+1 into the equation and solve for an by setting it equal to zero.

       

The surd is canceled once the whole equation is squared and this is left:

Now complete the square to solve for an:

To complete the square, take half of b, which is -1 in this case, and square it. Also make sure to add it to the other side of the equation because everything done to one side must be done to the other side.

After that, this is what is left:

This equation can be simplified by adding k and  , and then taking the square root of both sides to solve for an:

The negative value of this again must be disregarded because a surd cannot produce a negative answer

Therefore the exact value of this general infinite surd in terms of k is:

In order to check my work, I can substitute different values of k into the equation and see if they equal what they are supposed to. Since I already know the values for k=1 and k=2, I can test these two first:

For k=1:

This checks out, since in Part 1 I got the same expression for an

For k=2:

This also checks out, since in Part 2 I got the same expression for an

In order to find some values of k that would make the expression an integer, I would have to consider what numbers would make the  part of the equation a perfect square, and from those what numbers, when divided by two, would make the expression an integer.

So, to find values that would make k a perfect square, I would set:

, , ,  and so on…

Example Solution for k:

This table of values shows several the solutions for k using perfect square:

Because the expression   must be an integer, I will only use the values of 4k+1 that will make k an integer. These values are shown in bold in the table above.

Using those values, I can check to see if they will, in fact, make the expression , an integer:

This table of values shows the some of the other of the values of the expression, that I calculated by making a formula on Microsoft Excel:

The expression, , is an integer when k is an even, positive integer that will make the square root of 4k+1 a perfect square number. I also noticed that if I work backwards, there is a pattern:  

Let

This formula seems to work for all the values I have tried; for example:

Any of these values can be plugged back into the equation and check out like they’re supposed to.

If we test:

We will get:

Testing other Values of k:

Negative values would not be possible for k because this would make the square root negative and would give a non-real answer, so there is no reason to test negative numbers.

Any irrational number that is tested would not work either because an irrational number would not give an integer.

For example:

This is not an integer and therefore does not fit the requirement for the general statement.

Scope and/or Limitations of General Statement:

It would be difficult to test every single number, and that is a limitation in my theory that k=n(n-1). The answer to the infinite surd of one,

represents the only positive solution for the golden ratio. The golden ratio is an irrational number that goes on and on without any pattern in the numbers. This number appears many times in art, geometry, architecture, and other areas.

Other limitations for the general statement are that ……………

I arrived at my general statement by deciding what values of k would make the square root a perfect square, since a perfect square would give you an integer. First, I made a simple formula on Excel that would calculate the value of the expression,

If k equaled any integer, such as 1, 2, 3, 4, 5, 6, 7, 8, 9, 10...; I did this for every integer until 200. By using the technology in Excel, this task was fairly simple. After this, I noted all the values of k that would make the whole expression an integer, and then I made a table of these values. Using this strategy, I formulated my general statement based on the pattern that I noticed in the values of k.