Jonghyun Choe March 25 2011 Math IB SL Internal Assessment – LASCAP’S FractionThe goal of this task is to consider a set of fractions which are presented in a symmetrical,recurring sequence, and to find a general statement for the pattern.The presented pattern is: Row 1 Row 2 Row 3 Row 4 Row 5Step 1: This pattern is known as Lascap’s Fractions. En(r) will be used to represent the valuesinvolved in the pattern. r represents the element number, starting at r=0, and n representsthe row number starting at n=1. So for instance, , the second element on thefifth row. Additionally, N will represent the value of the numerator and D value of thedenominator. To begin with, it is clear that in order to obtain a general statement for the pattern,two different statements will be needed to combine to form one final statement. Thismeans that there will be two different statements, one that illustrates the numerators andanother the denominators, which will be come together to find the general statement. Tostart the initial pattern, the pattern is split into two different patterns; one demonstratingthe numerators and another denominators.
Step 2: This pattern demonstrates the pattern of the numerators. It is clear that all of thenumerators in the nth row are equal. For example all numerators in row 3 are 6. 1 1 3 3 3 6 6 6 6 10 10 10 10 10 15 15 15 15 15 15Row number (n) 1 2 3 4 5Numerator (N) 1 3 6 10 15N(n+1) - Nn N/A 2 3 4 5Table 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 numeratorincreases proportionally as the row number increases. It can be found that the value of N (n+1)- Nn increases proportionally as the sequence continues.The relationship between the row number and the numerator is graphically plotted and aquadratic fit determined, using loggerpro.Figure 1: The equation of the quadratic fit is the relationship between the numerator andthe row number. The equation for the fit is: N= or Equation 1
In this equation, N refers to the numerator. Therefore, N= or 0 is astatement that represents step 2 and also step 1.Step 3: In relation to table 1 and step 2, a pattern can be drawn. The difference between thenumerators of two consecutive rows is one more than the difference between the previousnumerators of two consecutive rows. This can be expressed in a formula N(n+1) - N(n) = N(n) -N(n-1) + 1. For instance, N(3+1) - N(3) = N(3) - N(2) +1.Using this method, numerator of 6th and 7th row can be determined. To find the 6th row’svalue, n should be plugged in as 5 so that N(6) can be found. As for the 7th row’s numerator, nshould be plugged in as 6.6th row numerator is therefore: N(5+1) - N(5) = N(5) - N(4)+1 N(6) – 15 = 15 – 10+1 N(6) = 15+6 N(6) = 217th row numerator is therefore: N(6+1) - N(6) = N(6) - N(5)+1 N(7) – 21 = 21 – 15 +1 N(6) = 42 – 15 + 1 N(6) = 28Not only by this method, but from the equation found in step 2, figure 1, 6th and 7th rownumerator can be found also.6th row numerator:7th row numerator: .5
Consequently, these are the values of numerators up to the 7th row. 1 1 3 3 3 6 6 6 6 10 10 10 10 10 15 15 15 15 15 15 21 21 21 21 21 21 21 28 28 28 28 28 28 28 28Using the method in step 3 and equation 1 in figure 1, it is evident that the numerator in the6th row is 21. Since both equations have brought same values, it can be concluded thatequation 1 is a valid statement that demonstrates the pattern of the numerator. Equation 1will be used later also, in order to form a general statement of the pattern of whole LACSAPFractions.Step 4: When examining the denominators in the LASCAP’S Fractions, their values are thehighest in the beginning, decreases, and then increases again. For example, thedenominators in row 5 are; 15 11 9 9 11 15. From this pattern, we can easily see that theequation for finding the denominator would be in a parabola form.Element 0 1 2 3 4 5Denominator 15 11 9 9 11 15 The relationship between the denominator and the element number is graphicallyplotted and a quadratic fit determined, using loggerpro.
Figure 2: This parabola describes the relationship between the denominator and elementnumber.The equation for the fit is : D = . In this equation, r refers to the elementnumber starting from 0, and being the first denominator value in the row. n refers to therow number starting from 1.To see if this equation work, the 3rd denominator value in the 3rd row was measured.D=D = 4 – 6 +6D=4With this equation, it is evident that the 6th and 7th row denominator values can be found.We already know the first and last denominators from when numerators were found; whichare 21 and 28.6th row second and sixth denominator: D = D= D = 166th row third and fifth denominator: D= D= D = 13
6th row fourth denominator: D= D= D = 127th row second and seventh denominator: D= D= D = 227th row third and sixth denominator: D= D= D = 187th row fourth and fifth denominator: D= D= D = 16Now, since the denominators in the 6th and 7th row are found, the sixth and seventh rowscan be drawn and added in the LACSAP’S Fractions.Consequently, these are the fractions up to the 7th row. 1 1 1 1 1 1 1 1 1 1 1 1 1 1
Now that the patterns for the LASCAP’S Fractions are found, all fractions can be expressed inthe form when it is the (r+1)th element in the nth row, starting with r=0.The general statement of the pattern is clearly found when using the equations for thenominator and the denominator. Therefore, the general statement for will be =In order to see if the equation works correctly, we can plug in number and figure out if thegeneral statement works out.For example, = = = = . Here, it is clear thatthe formula is applicable.In order to make sure that the general statement is valid, finding the additional rows of therecurring sequence of fractions by using the general statement above would be useful. Here,I chose to settle on 2 additional rows which are the 8th and 9th rows in the pattern.8th row numerator:9th row numerator:8th row second and eighth denominator: D = D= D = 298th row third and seventh denominator: D = D= D = 248th row fourth and sixth denominator: D = D= D = 218th row fifth denominator: D = D = D = 28
9th row second and ninth denominator: D = D= D = 379th row third and eighth denominator: D = D= D = 319th row fourth and seventh denominator: D = D= D = 279th row fifth and sixth denominator: D= D= D = 25Thus, these are the fractions up to the 9th row. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1This shows that the general statement for the symmetrical, recurring sequence of fractionsis valid and will continue to work.