an - an+1
as n gets very large.
lim(an - an+1) → 0
When n gets very large and approaches infinity, the value approaches 0 because the difference between these two values become very small.
This is a way to find the exact value of this infinite surd:
an+1 can also be written as an so therefore
an =
an2 = 1+an
an2 – an – 1 = 0
abc- formula:
a =
a1 = 1.618
a2 = -0.618
Since the value has to be a positive number, the exact value of this infinite surd is 1.618.
To research this furthermore, another example of an infinite surd can be considered:
Considering the infinite surd above, the first ten terms is:
b1 = = 1.847759065
b2 = = 1.961570561
b3 = = 1.990369453
b4 = = 1.997590912
b5 = = 1.999397637
b6 = = 1.999849404
b7 = = 1.999962351
b8 = = 1.999990588
b9 = = 1.999997647
By replacing an by bn the formula for this sequence is:
bn+1 =
because if we use b2 as an example, the answer will be:
b1+1 =
b2 =
b2 = 1.961570561
By plotting a graph, the relationship between n and bn can be shown:
As can be seen from this graph is that the value of bn increases as n gets larger. After a while the curve flattens out, but continue to rise. The values of bn approaches 2, but will actually never reach it. This can be understood by:
bn – bn+1
As n becomes very large, the value approaches 0 because:
lim(bn – bn+1) → 0
When n becomes very large and reaches infinity, the value approaches 0. This is because the difference between the two values is so small that the difference becomes insignificant.
The exact value of this infinite surd is:
bn+1 can also be written as bn so:
bn =
bn2 = 2 + bn
bn2 - bn – 2 = 0
abc- formula
b =
b1 = 2
b2 = -1
Since the value of bn can only be a positive number the exact value of this infinite surd will be 2.
Now, considering a general term
were the first term is:
The expression for the exact value of this general infinite surd in terms of k will be:
an2 = k + an
an2 – an – k = 0
abc formula
an =
an =
Above you can see the expression for the exact value for the general term because whatever number we use instead of k, we will find the exact value for the specific term. One example that will make the expression an integer is:
k = 2
an =
an =
an = 2
Another example is:
k = 6
an =
an =
an = 3
These values show that the answers of these values are integers. The general statement represented by k is:
an =
2an = 1+
(2an-1)2 = 1+4k
The expression above gives the solutions possible, with the answer expressed as an integer, for the possible k- values.
Testing the validity of the general statement using other values of k:
Ex. 1
k = -2
an =
an = No solution
Ex. 2
k = 0
an =
an =
an = 1
Ex. 3
k = 1.5
an =
an =
an = 1.82288
Ex. 4
k = 2
an =
an =
an = 2
Ex. 5
k = 2.1
an =
an =
an = 2.03297
Ex. 6
k = 2.5
an =
an =
an = 2.15831
As can be seen from example number 1 is that if k is a negative number we get square root of a negative number and therefore there will be no solution. Because of this, k cannot be a negative number.
From example 2 and 4 this can be found: There are few numbers that gives the answer of an integer in this sequence expressed with k. Some of the values are the numbers 0, 2, 6, 12 and 20 etc. From this we can see an increasing trend were:
0 = 2
2 = 2
6 = 2
12 = 2
20 = 2
We can see that from 0 to 2 the number we multiply by the constant, 2, increases by 1, and from 2 to 6 it increases by 2. From 6 to 12 it increases by 3, and from 12 to 20 it increases by 4.
If we use other numbers from 0 to 20 than those above, we get decimal numbers. This can be seen from example 3, 5 and 6. As k increases, the value of an increases as well.
The scope is 0< n >, because n cannot be a negative number.
