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# Continued Fractions

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Introduction

Continued Fractions

A continued fraction is any mathematical expression in the form of:  Where a0 is always and integer, and all other ‘ ’s such as a1, a2, and a3 are positive integers.  The number of terms can either be finite or infinite.  A more convenient way to denote continued fractions such as the one above would be to denote it by: Finite Continued Fractions

A finite continued fraction is an expression such as the one shown above which could end.  Every rational number can be equated to a finite continued fraction.  The only skill needed would be division of fractions.  Infinite Continued Fractions

Unlike the finite continued fractions, the chain of fractions never ends in an infinite continued fraction.  Every irrational number can be equated to an infinite continued fraction.  This fact was discovered and proven by the Swiss Mathematician, Leonhard Euler (1707-1783).  Some of Euler’s infinite continued fractions are as we will see below: A way to summarise this expression is to let denote the value of the continued fraction.  Usage of Continued Fractions

Continued fractions could be used to solve certain quadratic equations of the second degree.  Solving a quadratic equation using the

Middle

will continuously fluctuate but start to stabilize when =8.  Once reaches 23, will converge to a sustained value of 1.618033989.  This value will remain constant as long as  23

The same trend is brought up by the graph of and .  As the value of rises, will oscillate until reaches 25.  Only then will the value of converge to a consistent value of 0.  This value will stay the same as long as  25.

1.  When trying to determine the 200th term, we couldn’t obtain a negative value, only a positive value.  When the value of is still below 25, it oscillates between values which sometimes range from negative to positive values, but then as the value of increases it will converge to a constant specific positive value.  That means a negative value could only be obtained if 1. According to the table in question two, the exact value for the continued fraction is  1.

Conclusion

k were to be a negative number, no value could be obtained because a math error occurs and k must be an integer because that is the rule of a continued fraction;  if it weren’t an integer, though values could still be obtained unlike a negative k, it will no longer be considered a continued fraction.

Bibliography

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2. ## Lacsap's Fractions : Internal Assessment

By using the formula of 0.5n2 + 0.5n achieved from the method above, I entered it into the GDC. When the formula was entered into the GDC, the graph above was shown. This proves that it is a quadratic equation.

1. ## MATH Lacsap's Fractions IA

As the value of R2 of a certain model gets closer to 1, it becomes a better representative of the future data trends. So the closest R2 value to 1 will be the best model of the data. Table 2 Row Numerator 1 1 2 3 3 6 4 10

2. ## LACSAP Fractions. The aim of this portfolio is to discover an equation which ...

1 1 1 1 1 1 1 1 1 1 1 1 1 1 General Statement: To find the general statement for En (r), where the (r + 1)th element in the in the nth row. This is starting with r = 0 by combining the equation for the numerator and the denominator.

1. ## Lacsaps fractions are an arrangement of numbers that are symmetrically repeating based on a ...

hand side you have to multiply by 2 on the right hand side, to keep the equation balanced The rest will be presented in a table. Table 3 an illustration of the relationship between (n+1), n, k, 2k, N and 2N n n+1 N k 2k 2N 1 2 1

2. ## Lacsap's fractions - IB portfolio

is: 2.5×4=10 The 5th row can be calculated as: 0.5×5+0.5=3 and this is multiplied by the row number (5) is: 3×5=15 The results are the same as the original numerators so the formula seems to be working. Thus the general formula for kn is: kn=(0.5+0.5n)×(n+1)

1. ## Lascap's Fraction Portfolio

while L2 represents the value of numerator (N) Image 2 Caption: Image 2 shows the range set which is -2? n ? 6 for the x-axis while -2 ? y ? 20 for the y-axis Image 3 Caption: Image 3 shows the general equation obtained by using Quadratic Regression.

2. ## LAcsap fractions - it is clear that in order to obtain a general statement ...

1 2 3 4 5 Numerator (N) 1 3 6 10 15 N(n+1) - Nn N/A 2 3 4 5 Table 1: The increasing value of the numerators in relations to the row number. From the table above, we can see that there is a downward pattern, in which the numerator increases proportionally as the row number increases. • Over 160,000 pieces
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