. The values of an, are not exact because the values of an, have an infinite number of decimal points since it is a surd. When the value of n increases, the difference between two consecutive values of n are smaller than
Graph 1
Graph 1 exemplifies the direct relationship between an and n. As the value of n increases, the value of an increases. In other words, as n gets larger, the value of gets smaller. When n gets larger and gets closer to infinity, an increases at a slower rate (ex:). However, the difference between an and n will always be larger than 0 in terms of and can be represented by the equation an- an+1 > 0.
Taking another infinite surd sequence will further expand patterns. The following is the infinite surd of two:
…etc.
Figure 2. Relationship between n and bn
Table 2. Graph of infinite surd 2
The value of bn approaches 2, but never reaches it (Refer to Figure 2 and Table 2 above).
Considering a generalization of an infinite surd, variables can be used to replace an integer.
Let x be this value:
Because the equation above is not factorable, the quadratic equation is used:
1+4k has to be a perfect square because the Pythagorean Theorem states that is an irrational if n is not a perfect square. Note that the solution of x should be positive (Refer to Figure 1 and 2). Thus, ignore the optional subtraction sign because it will make x negative. The value of x is an integer; the numerator should be even because it has to be divisible by 2. Because is added to 1, the value of should be odd in order to make the whole value of the numerator even. Thus, 1+4k should be a perfect square so that is an integer.
To clarify the generalization above:
1+4k represents the variablein the quadratic equation. Then,
Consider that even; odd; b odd.
Let b = 2d+1
Then
Therefore, must be the product of two consecutive integers.
Then, which thus is an integer.
We can test the validity of this equation by solving for k in the quadratic formula:
Table 3
Figure 3
Figure 3 and Table 3 show further exemplify the equations and. There is direct relationship between k and x; when x increases, k increases. The limitations of the general equations are mainly based on the fact that x needs to be an integer. Thus 1+4k has to a perfect square and the square root should be odd in order for it to be divisible by two. The scope of k is infinite because the equation is based on a surd. The validity of the general statement of k only applies to even numbers that fit into the equation . Figure 3 and Table 3 represent that the value of an infinite surd is not always irrational and can be integer.
