Lacsap’s Fractions

Lacsaps Fractions IB Math 20 Portfolio By Lorenzo Ravani Lacsaps Fractions Lacsap is receding(prenominal) for dad. If we use Pascals triangle we can identify somas in Lacsaps fractions. The goal of this portfolio is to ? nd an comparability that describes the pattern presented in Lacsaps fraction. This equating must determine the numerator and the denominator for any run-in possible. Numerator Elements of the Pascals triangle form multiple horizontal lines (n) and diagonal tracks (r). The divisors of the ? rst diagonal row (r = 1) ar a analogue function of the row number n. For every other row, each gene is a parabolic function of n.Where r represents the constituent number and n represents the row number. The row numbers that represents the identical sets of numbers as the numerators in Lacsaps triangle, be the split second row (r = 2) and the seventh row (r = 7). These rows are respectively the third element in the triangle, and equal to each other because the triang le is symmetrical. In this portfolio we ordain cook up an equation for altogether these two rows to ? nd Lacsaps pattern. The equation for the numerator of the second and seventh row can be represented by the equation (1/2)n * (n+1) = Nn (r) When n represents the row number.And Nn(r) represents the numerator thereof the numerator of the sixth row is Nn(r) = (1/2)n * (n+1) Nn(r) = (1/2)6 * (6+1) Nn(r) = (3) * (7) Nn(r) = 21 escort 2 Lacsaps fractions. The numbers that are underlined are the numerators. Which are the homogeneous as the elements in the second and seventh row of Pascals triangle. Figure 1 Pascals triangle. The circled sets of numbers are the said(prenominal) as the numerators in Lacsaps fractions. Graphical Representation The plot of the pattern represents the relationship between numerator and row number. The graph goes up to the ninth row.The rows are represented on the x-axis, and the numerator on the y-axis. The plot forms a parabolic curve, representing an e xponential function increase of the numerator compared to the row number. Let Nn be the numerator of the intimate fraction of the nth row. The graph takes the shape of a parabola. The graph is parabolical and the equation is in the form Nn = an2 + bn + c The parabola passes through the points (0,0) (1,1) and (5,15) At (0,0) 0 = 0 + 0 + c At (1,1) 1 = a + b At (5,15) 15 = 25a + 5b 15 = 25a + 5(1 a) 15 = 25a + 5 5a 15 = 20a + 5 10 = 20a then c = 0 then b = 1 a restrict with other row numbers At (2,3) 3 = (1/2)n * (n+1) (1/2)(2) * (2+1) (1) * (3) N3 = (3) therefore a = (1/2) Hence b = (1/2) as well The equation for this graph therefore is Nn = (1/2)n2 + (1/2)n which simpli? es into Nn = (1/2)n * (n+1) Denominator The remainder between the numerator and the denominator of the same fraction will be the difference between the denominator of the current fraction and the previous fraction. Ex. If you take (6/4) the difference is 2. Therefore the difference betw een the previous denominator of (3/2) and (6/4) is 2. Figure 3 Lacsaps fractions showing differences between denominators Therefore the normal direction for ? nding the denominator of the (r+1)th element in the nth row is Dn (r) = (1/2)n * (n+1) r ( n r ) Where n represents the row number, r represents the the element number and Dn (r) represents the denominator. Let us use the manifestation we have obtained to ?nd the interior fractions in the sixth row. Finding the 6th row First denominator bet on denominator denominator = 6 ( 6/2 + 1/2 ) 1 ( 6 1 ) = 6 ( 3. 5 ) 1 ( 5 ) 21 5 = 16 denominator = 6 ( 6/2 + 1/2 ) 2 ( 6 2 ) = 6 ( 3. 5 ) 2 ( 4 ) = 21 8 = 13 -Third denominator Fourth denominator ordinal denominator denominator = 6 ( 6/2 + 1/2 ) 3 ( 6 3 ) = 6 ( 3. 5 ) 3 ( 3 ) = 21 9 = 12 denominator = 6 ( 6/2 + 1/2 ) 2 ( 6 2 ) = 6 ( 3. 5 ) 2 ( 4 ) = 21 8 = 13 denominator = 6 ( 6/2 + 1/2 ) 1 ( 6 1 ) = 6 ( 3. 5 ) 1 ( 5 ) = 21 5 = 16 We already know from the previous probe that the numerator is 21 for all interior fractions of the sixth row.Using these patterns, the elements of the 6th row are 1 (21/16) (21/13) (21/12) (21/13) (21/16) 1 Finding the 7th row First denominator Second denominator Third denominator Fourth denominator denominator = 7 ( 7/2 + 1/2 ) 1 ( 7 1 ) =7(4)1(6) = 28 6 = 22 denominator = 7 ( 7/2 + 1/2 ) 2 ( 7 2 ) =7(4)2(5) = 28 10 = 18 denominator = 7 ( 7/2 + 1/2 ) 3 ( 7 3 ) =7(4)3(4) = 28 12 = 16 denominator = 7 ( 7/2 + 1/2 ) 4 ( 7 3 ) =7(4)3(4) = 28 12 = 16 Fifth denominator Sixth denominator denominator = 7 ( 7/2 + 1/2 ) 2 ( 7 2 ) =7(4)2(5) = 28 10 = 18 denominator = 7 ( 7/2 + 1/2 ) 1 ( 7 1 ) =7(4)1(6) = 28 6 = 22 We already know from the previous investigation that the numerator is 28 for all interior fractions of th e seventh row. Using these patterns, the elements of the 7th row are 1 (28/22) (28/18) (28/16) (28/16) (28/18) (28/22) 1 General Statement To ? nd a general statement we combine the two equations needed to ? nd the numerator and to ? nd the denominator. Which are (1/2)n * (n+1) to ? d the numerator and (1/2)n * (n+1) n( r n) to ? nd the denominator. By letting En(r) be the ( r + 1 )th element in the nth row, the general statement is En(r) = (1/2)n * (n+1) / (1/2)n * (n+1) r( n r) Where n represents the row number and r represents the the element number. Limitations The 1 at the beginning and curio of each row is taken out before making calculations. Therefore the second element in each equation is now regarded as the ? rst element. Secondly, the r in the general statement should be greater than 0. thirdly the very ? rst row of the given pattern is counted as the 1st row.Lacsaps triangle is symmetrical like Pascals, therefore the elements on the left field side of the line of symmetry are the same as the elements on the right side of the line of symmetry, as shown in Figure 4. Fourthly, we only formulated equations based on the second and the seventh rows in Pascals triangle. These rows are the only ones that have the same pattern as Lacsaps fractions. Every other row creates either a linear equation or a different parabolic equation which doesnt match Lacsaps pattern. Lastly, all fractions should be kept when trim back provided that no fractions common to the numerator and the denominator are to be cancelled. ex. 6/4 cannot be reduced to 3/2 ) Figure 4 The triangle has the same fractions on both sides. The only fractions that occur only once are the ones get over by this line of symmetry. 1 Validity With this statement you can ? nd any fraction is Lacsaps pattern and to prove this I will use this equation to ? nd the elements of the 9th row. The subscript represents the 9th row, and the number in parentheses represents the element number. E9(1) First element E9(2) Second element E9(3) Third element n( n/2 + 1/2 ) / n( n/2 + 1/2 ) r( n r) 9( 9/2 + 1/2 ) / 9( 9/2 + 1/2 ) 1( 9 1) 9( 5 ) / 9( 5 ) 1( 8 ) 45 / 45 8 45 / 37 45/37 n( n/2 + 1/2 ) / n( n/2 + 1/2 ) r( n r) 9( 9/2 + 1/2 ) / 9( 9/2 + 1/2 ) 2( 9 2) 9( 5 ) / 9( 5 ) 2 ( 7 ) 45 / 45 14 45 / 31 45/31 n( n/2 + 1/2 ) / n( n/2 + 1/2 ) r( n r) 9( 9/2 + 1/2 ) / 9( 9/2 + 1/2 ) 3 ( 9 3) 9( 5 ) / 9( 5 ) 3( 6 ) 45 / 45 18 45 / 27 45/27 E9(4) Fourth element E9(4) Fifth element E9(3) Sixth element E9(2) Seventh element E9(1) Eighth element n( n/2 + 1/2 ) / n( n/2 + 1/2 ) r( n r) 9( 9/2 + 1/2 ) / 9( 9/2 + 1/2 ) 4( 9 4) 9( 5 ) / 9( 5 ) 4( 5 ) 45 / 45 20 45 / 25 45/25 n( n/2 + 1/2 ) / n( n/2 + 1/2 ) r( n r) 9( 9/2 + 1 /2 ) / 9( 9/2 + 1/2 ) 4( 9 4) 9( 5 ) / 9( 5 ) 4( 5 ) 45 / 45 20 45 / 25 45/25 n( n/2 + 1/2 ) / n( n/2 + 1/2 ) r( n r) 9( 9/2 + 1/2 ) / 9( 9/2 + 1/2 ) 3 ( 9 3) 9( 5 ) / 9( 5 ) 3( 6 ) 45 / 45 18 45 / 27 45/27 n( n/2 + 1/2 ) / n( n/2 + 1/2 ) r( n r) 9( 9/2 + 1/2 ) / 9( 9/2 + 1/2 ) 2( 9 2) 9( 5 ) / 9( 5 ) 2 ( 7 ) 45 / 45 14 45 / 31 45/31 n( n/2 + 1/2 ) / n( n/2 + 1/2 ) r( n r) 9( 9/2 + 1/2 ) / 9( 9/2 + 1/2 ) 1( 9 1) 9( 5 ) / 9( 5 ) 1( 8 ) 45 / 45 8 45 / 37 45/37 From these calculations, derived from the general statement the 9th row is 1 (45/37) (45/31) (45/27) (45/25) (45/25) (45/27) (45/31) (45/37) 1 Using the general statement we have obtained from Pascals triangle, and following the limitations stated, we will be subject to produce the elements of any given row in Lacsaps pattern. This equation determines the numerator and the denominator for every row possible.