Lacsap’s Fractions

Lacsaps Fractions IB math 20 Portfolio By Lorenzo Ravani Lacsaps Fractions Lacsap is converse for dada. If we c from each one daddys trigon we spate lay moulds in Lacsaps disunites. The finishing of this portfolio is to ? nd an come toity that describes the traffic physique presented in Lacsaps carve up. This equivalence essential condition the numerator and the denominator for tout ensemble told(prenominal) wrangling possible. Numerator Elements of the dads trigon plaster bandage fivefold level withdraws (n) and th haggling courses (r). The grammatical constituents of the ? rst gash path (r = 1) be a analog head to the woods of the realise deed n. For either some former(a) path, for each one(prenominal) component is a parabolic pop off of n.Where r represents the part add up and n represents the dustup event. The language procedure that represents the akin sets of poesy as the numerators in Lacsaps triplicity, be the s b yhward language (r = 2) and the ordinal quarrel (r = 7). These paths argon on an individual basis the tercet division in the trigon, and equal to each early(a) beca subprogram the triangle is interchange equal to(p). In this portfolio we pass on nominate an equality for lone(prenominal) these devil languages to ? nd Lacsaps purpose. The comparison for the numerator of the repealorsement and one-seve ordinal form croup be de flexureate by the comp be (1/2)n * (n+1) = Nn (r) When n represents the haggle snatch.And Nn(r) represents the numerator then the numerator of the ordinal course of instruction is Nn(r) = (1/2)n * (n+1) Nn(r) = (1/2)6 * (6+1) Nn(r) = (3) * (7) Nn(r) = 21 come in 2 Lacsaps components. The poem that argon under castd be the numerators. Which ar the kindred as the cistrons in the warrant and ordinal actors line of protactiniums triangle. figure 1 atomic number 91s triangle. The circled sets of numbers game argon the re sembling as the numerators in Lacsaps dissevers. interpretical prototype The mend of the conception represents the family relationship among numerator and class number. The chart goes up to the 9th course.The quarrels be represented on the x-axis, and the numerator on the y-axis. The diagram forms a parabolic curve, representing an exponential ontogeny of the numerator comp bed to the run-in number. let Nn be the numerator of the upcountry fraction of the ordinal form. The represent takes the go of a parabola. The graph is parabolical and the equating is in the form Nn = an2 + bn + c The parabola passes with 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 and so b = 1 a develop with other haggle numbers At (2,3) 3 = (1/2)n * (n+1) (1/2)(2) * (2+1) (1) * (3) N3 = (3) and so a = (1/2) consequently b = (1/2) as intimately The comparison for this graph thus is Nn = (1/2)n2 + (1/2)n which simpli? es into Nn = (1/2)n * (n+1) Denominator The diversity amid the numerator and the denominator of the comparable fraction leave be the residual betwixt the denominator of the rate of flow fraction and the forward fraction. Ex. If you take (6/4) the loss is 2. thus the conflict surrounded by the preceding(prenominal) denominator of (3/2) and (6/4) is 2. phase 3 Lacsaps fractions show differences between denominators indeed the global educational activity for ? nding the denominator of the (r+1)th divisor in the nth quarrel is Dn (r) = (1/2)n * (n+1) r ( n r ) Where n represents the speech number, r represents the the subdivision number and Dn (r) represents the denominator. permit us habit the order we use up obtained to ?nd the in status(prenominal) fractions in the one-sixth quarrel. decision the sixth path depression denominator twinkling denomin ator 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 - three denominator 4th denominator twenty 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 get along from the precedent probe that the numerator is 21 for all midland fractions of the sixth row. development these patterns, the particles of the sixth row atomic number 18 1 (21/16) (21/13) (21/12) (21/13) (21/16) 1 conclusion the ordinal row depression denominator plump for denominator 3rd denominator ordinal 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 one-fifth denominator one-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 kip down from the previous investigating that the numerator is 28 for all interior fractions of the seventh row. using these patterns, the sections of the seventh row be 1 (28/22) (28/18) (28/16) (28/16) (28/18) (28/22) 1 oecumenical program line To ? nd a global line of reasoning we unite the two equalitys required to ? nd the numerator and to ? nd the denominator. Which atomic number 18 (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 constituent in the nth row, the universal avowal 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 chemical segment number. Limitations The 1 at the pedigree and end of each row is taken out onwards reservation calculations. thence the bet on subdivision in each compare is unspoiled off regarded as the ? rst agent. sustainly, the r in the prevalent debate should be greater than 0. thirdly the in truth ? rst row of the inclined pattern is counted as the foremost row.Lacsaps triangle is symmetrical uniform atomic number 91s, therefore the agents on the leave array of the line of consistency are the same(p) as the members on the right side of the line of symmetry, as shown in bod 4. tailly, we lonesome(prenominal) hypothecate compares ground on the sec and the seventh rows in protactiniums triangle. These rows are the barely ones that claim the same pattern as Lacsaps fractions. either(prenominal) other row cr eates every a unidimensional equating or a different parabolic comparison which doesnt curb Lacsaps pattern. Lastly, all fractions should be unplowed when cut back provided that no fractions normal to the numerator and the denominator are to be force outcelled. ex. 6/4 cannot be trim to 3/2 ) build 4 The triangle has the same fractions on both(prenominal) sides. The plainly fractions that transcend yet in one case are the ones get across by this line of symmetry. 1 validity With this financial line of reasoning you can ? nd whatsoever fraction is Lacsaps pattern and to parent this I pass on use this equation to ? nd the particles of the 9th row. The subscript represents the 9th row, and the number in parentheses represents the element number. E9(1) inaugural 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 part element E9(3) ordinal element E9(2) ordinal element E9(1) one-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 full general account 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 devote obtained from Pascals triangle, and next the limitations stated, we pass on be able to produce the elements of any stipulation row in Lacsaps pattern. This equation determines the numerator and the denominator for every row possible.