Ozanalp Eryılmaz
11/B 138
13.02.2012
İHSAN DOĞRAMACI FOUNDATION
BİLKENT ERZURUM LABORATORY HIGH SCHOOL
MATHEMATICS STANDARD LEVEL PORTFOLIO ASSIGNMENT
INFINITE SURDS
Ozanalp ERYILMAZ
11/B
1
Ozanalp Eryılmaz
11/B 138
When we consider the expression √𝑘 + √𝑘 + √𝑘 + √𝑘 + √𝑘 + ⋯
13.02.2012
as an infinite surd and
we can see that for different values of k, results are all in a pattern. In this portfolio
assignment, I will find a general term for the exact value of the general infinite surd in terms
of k and justify my results using different examples.
The following expression is an example of an infinite surd.
√1 + √1 + √1 + √1 + √1 + ⋯
Consider this surd as a sequence of terms αn where:
α1 = √1 + √1
α2 = √1 + √1 + √1
α3 = √1 + √1 + √1 + √1
etc.
Find a formula for αn+1 in terms of αn:
By looking at the first 3 terms of the sequence we can write a general formula for αn+1 in
terms of αn:
αn+1 = √1 + 𝛼𝑛
Calculate the decimal values of the first 10 terms of the sequence:
α1 = √1 + √1 = 1, 41421
α2 = √1 + √1 + √1 = 1, 55377
α3 = √1 + √1 + √1 + √1 = 1, 59805
2
Ozanalp Eryılmaz
11/B 138
13.02.2012
α4 = √1 + √1 + √1 + √1 + √1 = 1, 61185
√
α5 = 1 + √1 + √1 + √1 + √1 + √1 = 1, 61612
√
α6 = 1 + 1 + √1 + √1 + √1 + √1 + √1 = 1, 61744
√
√
α7 = 1 + 1 + 1 + √1 + √1 + √1 + √1 + √1 = 1,61785
√
√
√
α8 = 1 + 1 + 1 + 1 + √1 + √1 + √1 + √1 + √1 = 1, 61798
√
√
√
√
α9 = 1 + 1 + 1 + 1 + 1 + √1 + √1 + √1 + √1 + √1 = 1, 61802
√
√
√
√
√
α10 = 1 + 1 + 1 + 1 + 1 + 1 + √1 + √1 + √1 + √1 + √1 = 1, 61803
√
√
√
√
√
etc.
Using technology, plot the relation between n and αn. Describe what you notice. What does
this suggest about the value of αn ‒ αn+1 as n gets very large? Use your results to find the exact
value for this infinite surd.
3
Ozanalp Eryılmaz
11/B 138
13.02.2012
Relation of an - n
1.65
1.6
an
1.55
1.5
Series1
1.45
1.4
0
2
4
6
8
10
12
n
As the value of n increases the difference between αn and αn+1 decreases. If n gets very large,
it means the difference will be 0.
Consider another infinite surd √2 + √2 + √2 + √2 + ⋯ where the first term is √2 + √2.
Repeat the entire process above to find the exact value for this surd.
b1 = √2 + √2 = 1, 84775907
b2 = √2 + √2 + √2 = 1,96157056
b3 = √2 + √2 + √2 + √2 = 1, 99036945
b4 = √2 + √2 + √2 + √2 + √2 = 1, 99759091
√
b5 = 2 + √2 + √2 + √2 + √2 + √2 = 1, 99939764
4
Ozanalp Eryılmaz
11/B 138
13.02.2012
√
b6 = 2 + 2 + √2 + √2 + √2 + √2 + √2 = 1, 99984940
√
√
b7 = 2 + 2 + 2 + √2 + √2 + √2 + √2 + √2 = 1, 99996535
√
√
√
b8 = 2 + 2 + 2 + 2 + √2 + √2 + √2 + √2 + √2 = 1, 99999059
√
√
√
√
b9 = 2 + 2 + 2 + 2 + 2 + √2 + √2 + √2 + √2 + √2 = 1, 99999765
√
√
√
√
√
b10 = 2 + 2 + 2 + 2 + 2 + 2 + √2 + √2 + √2 + √2 + √2 = 1, 99999941
√
√
√
√
√
etc.
5
Ozanalp Eryılmaz
11/B 138
13.02.2012
bn
Relation of bn-n
2.02
2
1.98
1.96
1.94
1.92
1.9
1.88
1.86
1.84
Series1
0
2
4
6
8
10
12
n
Same with the first graph, the difference between αn and αn+1 will be zero. For this graph,
result of the infinite surd approaches 2.
Now consider the general infinite surd √𝑘 + √𝑘 + √𝑘 + √𝑘 + ⋯ where the first term is
√𝑘 + √𝑘. Find an expression for the exact value of this general infinite surd in terms of k.
c1 = √𝑘 + √𝑘
x = √𝑘 + √𝑘 + √𝑘 + √𝑘 + ⋯
x2 = 𝑘 + √𝑘 + √𝑘 + √𝑘 + ⋯
x2 = 𝑘 + x
x2 ‒ x ‒ k = 0
x2 ‒ x = k
x (x ‒ 1) = k
The value of an infinite surd is not always an integer.
Find some values of k that make the expression an integer. Find the general statement that
represents all the values of k for which the expression is an integer.
6
Ozanalp Eryılmaz
11/B 138
13.02.2012
Since the range of the equation represents consecutive numbers, Every consecutive number
starting with x = 1 satisfies the equation and makes the expression an integer. For example for
k = 12, x = 4. This means 12 for k makes the expression an integer which is 4.
Test the validity of your general statement using other values of k.
For example;
k = 12
x = 4. This means 12 for k makes the expression an integer which is 4.
Discuss the scope and/or limitations of your general statement.
The general formula is not appliable for negative numbers, because when two negative
numbers are multiplied, the result will become positive and the result of infinite surd will be
negative which is impossible.
Explain how you arrived at your general statement.
Firstly, we found a general formula for αn+1 in terms of αn. We used two examples; the first
one is with the number 1 and the second one is with 2. We graphed them and reached a
general statement which is as the value of n gets bigger, the difference between αn and αn+1
decreases and approaches zero. Then we used a general formula in terms of k. We equaled it
to x, and formed an equation. From this equation we found out the integer values of the
infinite surd.
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