a2 can be written as
Therefore, an+1 =
- The first ten terms of the sequence
-
Plot the relation between n and an
→ I can see from the graph above that the value of an approaches approximately 1.618 but never reach it. Also, I can suggest from the chart below that the consecutive differences are rapidly approaching zero as n gets larger.
-
What does this suggest about the value of an+1 – an as n gets very large?
- Use your results to find the exact value for this infinite surd
Let an be S
S =
S2 = 1 + S
S2 – S - 1 = 0
Use quadratic formula.
Therefore, S =
The value must be positive since the graph showed that there are no negative values. So, disregard the negative sign.
Thus, the exact value is
-
Consider another infinite surd where the first term is.
A1 =
A2 =
A3 = and so on.
-
Find a formula for an+1 in terms of an.
Since a1 = and a2 =,
A2 can be written as
Therefore, an+1 =
- The first ten terms of the sequence.
-
Plot the relation between n and an
→ I can see easily from the graph that the value of an approaches 2 but never reach it and the graph becomes less steep and the value of an does not increase as the value of n increases.
-
What does this suggest about the value of an+1 – an as n gets very large?
→ As you can see from the table on the left, the differences between an+1 and an become smaller and rapidly approaching zero as n gets larger.
- Find the exact value for this infinite surd.
* Let an be X
X =
X2 = 2 + X
X2 – X – 2 = 0
Use quadratic formula.
Therefore, X =
The value must be positive since the graph showed that there are no negative values, so disregard the negative sign.
Thus, X = = 2
- Finding an expression for the exact value of following general infinite surd in terms k.
The general infinite surd where the first term is.
-
Consider this surd as a sequence of terms bn
B1 =
B2 =
B3 = and so on.
-
Find a formula for bn+1 in terms of bn.
Since b1 = and b2 =,
B2 can be written as
Therefore, bn+1 =
- The first ten terms of the sequence
→ From the calculation for the differences between an+1 and an above, I can know that the difference between an+1 and an become smaller and rapidly approaching zero as n gets larger.
Therefore, bn+1 – bn = 0
- Find the expression for the exact value of this general infinite surd.
bn+1 – bn = 0
bn+1 = bn
Substitute for bn+1 because the formula for bn+1 =
So, = bn
* Let bn be X
X =
X2 = K + X
X2 –X –K = 0
Use the quadratic formula.
X = =
- The value of an infinite surd is not always an integer.
-
Find some values of k that make the expression an integer.
So, it is known that the expression for the exact value of
= (X =)
X =
If X is an integer, the numerator has to be even because it is divided by 2. So has to be odd which means that 1+4k has to be also odd.
Therefore, 1+4k has to be a perfect square so that is an integer.
Example 1) If I put 12 into the k,
X = = 4
Example 2) If I put 2 into the k,
X= = 2
Example 3) If I put 6 into the k,
X= = -2
So, some values of k that make the expression an integer are 2, 6 and 12.
-
Find the general statement that represents all the values of k for which the expression is an integer.
Sinceis an odd perfect square, let m be any odd number
Hence, 1+4k = (m) 2 because the square of an odd number is also an odd number.
1+4k = m2
4k = m2 -1
Thus, k =
-
Test the validity of general statement using other values of k
The general statement: k =
Testing 1) If k = -5
-5 =
-20 = m2-1
m2 = -19
So, there is no value of m.
Thus, k cannot be negative.
Testing 2) If k =
=
3 = m2-1
m2 = 4
m = 2
Since X =,
X = = or - So, X is not an integer.
Thus, k cannot be a fraction.
Testing 3) If k = 0
X = = 1 or 0 So, X is an integer.
Thus, k has to be an integer that is greater or equal to 0.
- Discuss the scope and/or limitations of your general statement.
- According to the three tests above, I found that k cannot be a negative number and a fraction. If k is a negative number and a fraction, X, which expresses the exact value of the general infinite surd, cannot be an integer. It becomes an integer only when k is an integer that is greater or equal to 0.
- Explain how you arrived at your general statement.
- First, I found a formula for the general infinite surd where the first term is. I considered this surd as a sequence of terms bn, so I could find that bn+1 =. Then, I could know from the calculation for the differences between an+1 and an that the differences between bn+1 and bn become smaller and rapidly approaching zero as n gets larger. After that, I substituted for bn+1 because the formula for bn+1 = and I got = bn. I let bn be X, so I got X2 –X –K = 0 at the end. And then I used the quadratic formula to find the expression for the exact value of this general infinite surd which is. After I got the expression for the exact value, I tried to find some values of k that make the expression an integer and I could realize that if X is an integer, the numerator has to be even because it is divided by 2. So, I found that has to be an odd perfect square to X be an integer. Since I found out this fact, I let m be any odd number thus, 1+4k = (m)2 because the square of an odd number is also an odd number. Finally, I developed the equation and I arrived at my general statement which is k = .
