To find what an is, one must perceive an and an+1 as being the same variables.
an+1 = ← Get rid of the subscripts.
(a = 2← Square it to get rid of the radical sign.
a2 = 1 + a← Get all numbers on one side. Then set theme equal to 0.
a2 – a – 1 = 0← One now uses the quadratic formula because this is unable to be factored.
→ → →
The asymptote is approximately 1.618. This makes sense when one looks at the data. Since the graph only shows numbers above 0, there are no negative numbers. Therefore, one must only look for the positive answer. One can also see that there is a limit for our “a” value. One can take this process when solving any other infinite surd, example:
…
When one sees this, one can generate a pattern
b1 =
b2 =
b3 =
Using this pattern, one can find the next 10 consecutive terms.
a1 1.847759065
a2 1.961570561
a3 1.990369453
a4 1.997590912
a5 1.999397637
a6 1.999849404
a7 1.999984940
a8 1.999990588
a9 1.999997647
a10 1.999999412
One can also observe that this graph also reaches an asymptote. This means that an approaches 2, but never quite reaches it. The formula that one can see is an+1 = n. Again, one must find the asymptote:
an+1 = ← Get rid of the subscripts.
(a = 2← Square it to get rid of the radical sign.
a2 = 2 + a← Get all numbers on one side. Then set them equal to 0.
a2 – a – 2 = 0← One now uses the quadratic formula because this is unable to be factored.
→ → →
The asymptote is 2. This makes sense when one looks at the data. Since the graph only shows numbers above 0, there are no negative numbers. Therefore, one must only look for the positive answer. The graph eventually will get to 2. One can take this process as well when solving a general example of an infinite surd. For example:
Using the methods discovered in the previous infinite surds, one can generate a general formula: an+1 = n.
The process for simplifying this is not complex at all. One only applies the method so far.
an+1 = ← Get rid of the subscripts.
(a = 2← Square it to get rid of the radical sign.
a2 = k + a← Get all numbers on one side. Then set them equal to 0.
a2 – a – k = 0← One now uses the quadratic formula because this is unable to be factored.
→ → →
We only use the positive values for “a” because we have observed this throughout the pattern. The general statement that represents the values of k for when the expression is an integer is. The reason for which one does not use the other answer to this formula is because a negative number is a nonexistent number, in our case. In other words, €R. Now, one must find that values of k that will make an infinite surd equal to integers. When we look back at the infinite surd of 2, we can see that when k = 2, the infinite surd is equal to 2. To help analyze this, one must think about what 4k means. According to 1+ 4k, 4k will always give us a positive number alone (4·2 = 8, 4·5=20, 4·8 = 32), however if we add one to this then we will always get a negative number (8+1 = 9, 20 + 1 = 21, 32 + 1 = 33). is odd if 4k+1 is a perfect square. Now we can substitute integers and see what new patterns can be generated.
4 = →8 = 1 + → 7 =→49 = 1+4k→ 48 = 4k→ k = 12
3 = →6 = 1 + → 5=→25 = 1+4k→ 24 = 4k→ k = 6
5 = →10 = 1 + → 9 =→81 = 1+4k→ 80 = 4k→ k = 20
A general statement evolved from this is to use the value of k (2) and multiply it by the surd than add it to the “k” value found and you will find the next consecutive term.
Example:
(4(2)) + 12 = 20
(3(2)) + 6 = 12
Therefore we can predict that
(5(2)) + 20 = 30
When 6 is the infinite surd, k will be 30.
This statement however does have a limitation, it being that this statement needs a value of k to find the next consecutive term. However if using the formula, we can get k and then apply this generalization. This generalization is just a pattern formulated as more examples were being created and developed. This is why when solving such a complex mathematical problem, one must always work it out and break it down where necessary.
