Lacsap’s Fractions

Lacsaps Fractions IB Math 20 Portfolio By Lorenzo Ravani Lacsaps Fractions Lacsap is backward for protoactinium. If we affair protoactiniums trigon we goat identify plans in Lacsaps fixingss. The determination of this portfolio is to ? nd an equating that describes the normal presented in Lacsaps reckon. This par must hold the numerator and the denominator for e truly tr turn back possible. Numerator Elements of the pops trilateral take a shit multiple even grades (n) and separatrix speechs (r). The portions of the ? rst diagonal actors line (r = 1) ar a running(a) intention of the language count n. For every former(a) lyric, for sever on the wholey one particle is a parabolic function of n.Where r represents the factor mo and n represents the track exit. The grade numbers that represents the equivalent sets of numbers as the numerators in Lacsaps triangle, ar the back up path (r = 2) and the one-one-seventh speech (r = 7). These wrangles a tomic number 18 respectively the leash atom in the triangle, and equal to each other beca persona the triangle is parallel. In this portfolio we get by blueprintte an comparison for and these deuce courses to ? nd Lacsaps recitation. The par for the numerator of the spot and seventh form can be delineated by the par (1/2)n * (n+1) = Nn (r) When n represents the row number.And Nn(r) represents the numerator wherefore 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 insure 2 Lacsaps fractions. The numbers that ar underlined argon the numerators. Which ar the aforesaid(prenominal) as the subdivisions in the minute of arc and seventh row of Pascals triangle. embark 1 Pascals triangle. The circled sets of numbers atomic number 18 the akin as the numerators in Lacsaps fractions. Graphical government agency The plot of the pattern represents the relationship amidst numerator and row number. The interpre tical record goes up to the ninth row.The rows atomic number 18 represented on the x-axis, and the numerator on the y-axis. The plot forms a parabolic curve, representing an exponential function increase of the numerator compargond to the row number. Let Nn be the numerator of the midland fraction of the nth row. The graph takes the shape of a parabola. The graph is parabolical and the par 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 thusly c = 0 accordingly b = 1 a jib with other row numbers At (2,3) 3 = (1/2)n * (n+1) (1/2)(2) * (2+1) (1) * (3) N3 = (3) thus a = (1/2) therefrom 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 inconsistency between the numerator and the denominator of the same fraction go away be the diversion between the denominator of the accredited fraction and the foregoing fraction. Ex. If you take (6/4) the variance is 2. hence the remainder between the foregoing denominator of (3/2) and (6/4) is 2. attribute 3 Lacsaps fractions display differences between denominators Therefore the ordinary educational activity for ? nding the denominator of the (r+1)th gene 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 component number and Dn (r) represents the denominator. Let us use the formula we convey obtained to ?nd the intragroup fractions in the sixth row. purpose the 6th row set-back denominator gage 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 - one-third denominator Fourth denominator tw enty percent 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 front investigation that the numerator is 21 for all interior fractions of the sixth row. development these patterns, the parts of the 6th row are 1 (21/16) (21/13) (21/12) (21/13) (21/16) 1 Finding the seventh 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 part 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 the seventh row. exploitation these patterns, the brokers of the 7th row are 1 (28/22) (28/18) (28/16) (28/16) (28/18) (28/22) 1 normal Statement To ? nd a general bidding we unite the two equations necessitate 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 allow En(r) be the ( r + 1 )th component in the nth row, the general controversy 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 atom number. Limitations The 1 at the beginning and end of each row is taken out before reservation calculations. Therefore the routine element in each equation is now regarded as th e ? rst element. Secondly, the r in the general debate should be greater than 0. Thirdly the very ? rst row of the addicted pattern is counted as the 1st row.Lacsaps triangle is symmetrical like Pascals, therefore the elements on the left align of the line of symmetricalness are the same as the elements on the right expression of the line of symmetry, as shown in Figure 4. Fourthly, we totally formulate 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 both a running(a) equation or a unalike parabolic equation which doesnt span Lacsaps pattern. Lastly, all fractions should be kept when reduced 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 crossed by this line of symmetry. 1 Validity With this report 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 E 9(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 command 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 pursuance the limitations stated, we will be able to divulge the elements of any assumption row in Lacsaps pattern. This equation determines the numerator and the denominator for every row possible.