a = 1 + 5
2
In decimal form:
a ≈ 1.618033989
This number completes the criterion of the graph, as it is between 1.6 and 1.7, directly where the asymptote should be. Also, the decimal value of a10≈ 1.6180285974702, which is shown by the graph, is very close to the asymptote. The difference between the approximate value for the infinite surd and a10 is only 5.3912797∙10-6, or 0.0000053912797, which is very precise and means that the asymptote is very near a10, like the graph predicts. If this value is also added to the graph, it will show that it is very near where the asymptote belongs:
The following graph shows the asymptote added unto the preliminary graph given on page 3:
To find the exact value of another infinite surd:
2+ 2 + 2 + 2 + …
The process is very similar. By observing the progression of the surds:
a1 = 2 + 2
a2 = 2 + 2 + 2
a3 = 2 + 2 + 2 + 2
etc.
The exact same pattern can be observed with this surd as the previous, with the obvious exception of the common difference, and therefore the formulas are nearly identical.
an+1 = 2 + an
Using the same method, the first 10 decimal values of this sequence are:
a1 ≈ 1.8477590650226
a2 ≈ 1.9615705608065
a3 ≈ 1.9903694533444
a4 ≈ 1.9975909124103
a5 ≈ 1.9993976373924
a6 ≈ 1.9998494036783
a7 ≈ 1.9999623505652
a8 ≈ 1.9999905876192
a9 ≈ 1.9999976469034
a10≈ 1.9999994117258
These values also appear to be approaching an asymptote. In order to find the value of the asymptote, it the formula needs to be simplified like the previous formula.
an+1 = 2 + an
Assign variable a to an+1 and an.
( a = 2 + a )2
a2 = 2 + a
a2 – a – 2 = 0
Again this can be simplified further using the quadratic equation.
a = 1
b = -1
c = -2
a = 1± (-1)2–4(1)(-2)
2(1)
a = 1 ± 9
2
a = 2, -1
This data is also all positive, and since there are no negatives that show up on the graph, the answer is undoubtedly 2.
The graph supports this. As the graph approaches 2 as can be seen by
a10≈ 1.9999994117258
By adding the horizontal line y=2 into the graph, the asymptote can be viewed along side the data points, giving a visual representation to the answer described above.
The only difference in the formulas of the infinite surd with the square root of 1 and the second infinite surd with the square root of 2 was that the first number inside the square root matched the number inside the infinite surd for each consecutive term. Therefore it can be predicted (under the circumstances that the surd has variable inside of it rather than a number) that that variable should go in the place of the number. Therefore the formula is represented by:
an+1 = k+ an
This can be simplified in the exact same manner as the previous formulas.
an+1 = k + an
Assign variable a to an+1 and an.
( a = k + a )2
a2 = k + a
a2 – a – k = 0
This can be simplified further using the quadratic equation:
a = 1
b = -1
c = -k
a = 1 ± (-1)2–4(1)(-k)
2(1)
a = 1 ± 1+4k
2
The pattern dictates that the answer must be positive, unless k is equal to a negative number, which is an impossibility due to the fact that the square root of a negative number does not exist, further simplifying the equation to:
a = 1 + 1+4k
2
By comparing several a10 values to their respective values of k according to the formula above, there should be an almost nonexistent difference, indicating that the formula above is correct
The differences are all very small because the line is constantly approaching the asymptote. The difference will never actually reach zero however, because by definition, a line never actually reaches an asymptote. This very small difference indicates that this equation does in fact represent the asymptote.
By working the problem in reverse, it becomes possible to calculate which values of k in an infinite surd are equal to integers.
According to the calculations above, it is already known that when k=2, the value of the infinite surd is also equal to 2.
(3 = 1 + 1+4k ) 2
2
(6 =1+ 1+4k )-1
(5 = 1+4k ) 2
(25 = 1+4k)-1
(24 = 4k)/4
6 = k
According to the calculations above, when k=6 the value of the infinite surd is 3. Using the same method as above, (albeit some guess and check) other values include:
When k=12, the value of the infinite surd is 4
When k=20, the value of the infinite surd is 5
When k=30, the value of the infinite surd is 6
When k=42, the value of the infinite surd is 7
The general statement that represents all the values of k for which the expression is an integer can be found by analyzing the values above. For example, when k=42, the value of the surd is 7. 42 is a multiple of 7: 42/7=6. Some possible statements from this would be that in order to find k, the value of the surd must be multiplied by 6. Another possible statement could be that in order to find k, one must multiply the value of the surd by the value of one less than that same surd. A final statement could be that if the k value at 6 has already been deduced, than the surd is simply two times the value of that previous surd with an integral value of the surd added unto the value of the previous surd (All of these statements were deduced by active analysis of the numbers above).
In order for one or more of these statements to be true, they must be tested with different numbers to determine the validity of each.
The first statement deduced that k is equal to the value of the surd times 6, or that the value of the surd is equal to the value of k/6. Using deductive reasoning, when the value of the surd is 6, than k should be equal to 36. It is not however in actuality being 30. This statement can thus be verified as untrue
The second statement states that the value of k is equal to the sum of the value of surd and the value of the surd subtracted by 1, represented formulaically by:
k= an + an-1
According to this statement, when the value of the surd is 6, k should be 30. From the testing of the first statement, this is already known to be true. The mathematical proof using this derived formula is this: (6)(6-1)=30. In addition, according to this statement, when the value of the surd is 5, k should be equal to 20, because (5)(5-1)=20.
Proof:
(5 = 1 + 1+4k ) 2
2
(10=1+ 1+4k )-1
(9 = 1+4k ) 2
(81 = 1+4k)-1
(80 = 4k)/4
20 = k
This is also true, so this statement is true in these low cases, but it may not be in higher numbers. According to the formula for the general infinite surd, when the value of surd is 11,249, that k is 126,528,752.
Proof:
(11,249 = 1 + 1+4k ) 2
2
(22,498 =1+ 1+4k )-1
(22,497 = 1+4k ) 2
(506,115,009 = 1+4k)-1
(506,115,008= 4k)/4
126,528,752= k
According to this statement, when the value of the surd is 11,249, k is 126, 528, 752, because (11,249)(11,249-1)=126,528,752. This shows that this statement is true for numbers of both large and small values.
Even though statement two appears to work for all integers regardless of value, it still has its limitations. The statement can only kind the value of k, it cannot find what the value of the surd is with a given k value. This is an acceptable limitation because the formula:
an+1 = (k+ an)1/2 . Other than this rather general and unimportant limitation, the equation should work regardless of the numerical values plugged into it, as both high and low values were used in the testing of the validity of the statement. Thus, in order to find the value of k when the value of the surd is equal to some integer, simply multiply that integer by the same integer subtracted by one.
I arrived at this general statement through the help of the internet and the use of deductive reasoning. Using the website Math Help Forum, I saw some examples of some basic solutions of infinite surds. From these basic solutions, through intense deductive reasoning, I arrived at the general statement arrived before. It was essentially guess and check until I found a formula that worked. I came up with several other equations to represent this general statement, but unfortunately none of them worked. The statement above is the only one I could feasibly find that worked under every circumstance I threw at it.
Sources:
“Quadratic Regression” (2001). Available from: Macon State College
(Accessed December 3rd 2008
“Infinite Surds” (2000). Available from: vBulletin
(Accessed December 2nd 2008).
