// fltk 7Fl_Tile 36 -191 901 62 20 [ // fltk N4xcas23Comment_Multiline_InputE 36 -191 901 61 20 On construit une programme itératif de paramètres l'indice n, les valeurs de u(0) et des paramètres a et b.£On a u(n)=u(n-1)+(k-1)*a+b donc val:=normal(val+(k-1)*a+b) , // fltk N4xcas10Log_OutputE 36 -130 901 1 20 ] , // fltk 7Fl_Tile 36 -127 901 313 20 [ // fltk N4xcas7EditeurE 36 -127 901 226 20 141 , u(n,u0,a,b):={ local val,k; if (n==0) return u0; val:=u0; for (k:=1;k<=n;k++) { val:=normal(val+(k-1)*a+b); } return val; }:;, // fltk N4xcas10Log_OutputE 36 99 901 44 20 // Parsing u£// Sucess compiling u£ , // fltk N4xcas8EquationE 36 143 901 43 16 "Done" ] , // fltk 7Fl_Tile 36 188 901 25 20 [ // fltk N4xcas23Comment_Multiline_InputE 36 188 901 24 20 Par exemple, pour a=b=u(0)=1, on obtient pour u10 , // fltk N4xcas10Log_OutputE 36 212 901 1 20 ] , // fltk 7Fl_Tile 36 215 901 68 20 [ // fltk N4xcas19Multiline_Input_tabE 36 215 901 24 20 u(10,1,1,1) , // fltk N4xcas10Log_OutputE 36 239 901 1 20 , // fltk N4xcas8EquationE 36 240 901 43 16 56 ] , // fltk 7Fl_Tile 36 285 901 41 20 [ // fltk N4xcas23Comment_Multiline_InputE 36 285 901 40 20 Représentons graphiquement les premiers termes de la suite. On crée une séquence S de coordonnées, vide au départ, et à laquelle£ on ajoute les points coordonnées (k,u(k)) pour k variant de 0 à n. , // fltk N4xcas10Log_OutputE 36 325 901 1 20 ] , // fltk 7Fl_Tile 36 328 901 305 20 [ // fltk N4xcas7EditeurE 36 328 901 218 20 150 , AffSuite(n,u0,a,b):={ local S,k; S:=NULL; affichage(point_carre+rouge); for(k:=0;k<=n;k++){ S:=S,point(k,u(k,u0,a,b)); } return S; }:;, // fltk N4xcas10Log_OutputE 36 546 901 44 20 // Parsing AffSuite£// Warning: u declared as global variable(s) compiling AffSuite£ , // fltk N4xcas8EquationE 36 590 901 43 16 "Done" ] , // fltk 7Fl_Tile 36 635 901 325 20 [ // fltk N4xcas19Multiline_Input_tabE 36 635 901 24 20 AffSuite(10,0,1,1) , // fltk N4xcas10Log_OutputE 36 659 901 1 20 , // fltk 7Fl_Tile 36 660 901 300 20 [ // fltk N4xcas7Graph2dE 36 660 781 300 20 0,10,0,55,[pnt(pnt[0,100663297]),pnt(pnt[1+i,100663297]),pnt(pnt[2+3*i,100663297]),pnt(pnt[3+6*i,100663297]),pnt(pnt[4+10*i,100663297]),pnt(pnt[5+15*i,100663297]),pnt(pnt[6+21*i,100663297]),pnt(pnt[7+28*i,100663297]),pnt(pnt[8+36*i,100663297]),pnt(pnt[9+45*i,100663297]),pnt(pnt[10+55*i,100663297])],-5,5,1,0,0,0,1,5,1,0,1,5,0 , // fltk 7Fl_Tile 817 660 120 300 20 [ // fltk N4xcas14Mouse_PositionE 817 660 120 40 20 [] , // fltk 8Fl_Group 817 700 120 115 20 [ // fltk 9Fl_Button 817 700 40 23 20 [] , // fltk 9Fl_Button 857 700 40 23 20 [] , // fltk 9Fl_Button 897 700 40 23 20 [] , // fltk 9Fl_Button 817 723 40 23 20 [] , // fltk 9Fl_Button 857 723 40 23 20 [] , // fltk 9Fl_Button 897 723 40 23 20 [] , // fltk 9Fl_Button 817 746 40 23 20 [] , // fltk 9Fl_Button 857 746 40 23 20 [] , // fltk 9Fl_Button 897 746 40 23 20 [] , // fltk 9Fl_Button 817 769 40 23 20 [] , // fltk 9Fl_Button 857 769 40 23 20 [] , // fltk 9Fl_Button 897 769 40 23 20 [] , // fltk 9Fl_Button 817 792 40 23 20 [] , // fltk 11Fl_Menu_Bar 857 792 80 23 20 [] ] , // fltk 8Fl_Group 817 815 120 145 20 [ ] ] ] ] , // fltk 7Fl_Tile 36 962 901 236 20 [ // fltk N4xcas23Comment_Multiline_InputE 36 962 901 235 20 Le graphique nous laisse penser que le progression est assez régulière, voire que le graphe ressemble à une £parabole. On essaie donc de rechercher une expression de u(n) sous la forme£ u(n)=A.n^2+B.n+C£Pour trouver A, B et C, on utilise trois valeurs particulières de la suite (autant que d'inconnues) pour avoir un £système linéaire de trois équations à trois inconnues qu'on résout avec £linsolve([eq1,eq2,eq3],[inconnue1,inconnue2,inconnue3]).£Pour gagner du temps, on utilise la fonction "$" £ ( formule(n))$(1ere valeur de n..derniere valeur de n))£qui répète une formule dépendant de n pour n, entier, variant de 1ere valeur à dernière valeur. , // fltk N4xcas10Log_OutputE 36 1197 901 1 20 ] , // fltk 7Fl_Tile 36 1200 901 92 20 [ // fltk N4xcas19Multiline_Input_tabE 36 1200 901 24 20 linsolve([(A*n^2+B*n+C=u(n,0,1,1))$(n=0..2)],[A,B,C]) , // fltk N4xcas10Log_OutputE 36 1224 901 1 20 , // fltk N4xcas8EquationE 36 1225 901 67 16 [1/2,1/2,0] ] , // fltk 7Fl_Tile 36 1294 901 25 20 [ // fltk N4xcas23Comment_Multiline_InputE 36 1294 901 24 20 Comparons donc le tracé précédent avec la courbe d'équation y=(1/2)x^2+(3/2)x , // fltk N4xcas10Log_OutputE 36 1318 901 1 20 ] , // fltk 7Fl_Tile 36 1321 901 325 20 [ // fltk N4xcas19Multiline_Input_tabE 36 1321 901 24 20 AffSuite(10,0,1,1),plotfunc((1/2)*x^2+(1/2)*x,x=0..10) , // fltk N4xcas10Log_OutputE 36 1345 901 1 20 , // fltk 7Fl_Tile 36 1346 901 300 20 [ // fltk N4xcas7Graph2dE 36 1346 781 300 20 0,10,0,55,[pnt(pnt[0,100663297]),pnt(pnt[1+i,100663297]),pnt(pnt[2+3*i,100663297]),pnt(pnt[3+6*i,100663297]),pnt(pnt[4+10*i,100663297]),pnt(pnt[5+15*i,100663297]),pnt(pnt[6+21*i,100663297]),pnt(pnt[7+28*i,100663297]),pnt(pnt[8+36*i,100663297]),pnt(pnt[9+45*i,100663297]),pnt(pnt[10+55*i,100663297]),pnt(pnt[curve(group[pnt[x+(i)*(1/2*x^2+1/2*x),x,0.0,10.0],group[0.0,0.025+0.0128125*i,0.05+0.02625*i,0.075+0.0403125*i,0.1+0.055*i,0.125+0.0703125*i,0.15+0.08625*i,0.175+0.1028125*i,0.2+0.12*i,0.225+0.1378125*i,0.25+0.15625*i,0.275+0.1753125*i,0.3+0.195*i,0.325+0.2153125*i,0.35+0.23625*i,0.375+0.2578125*i,0.4+0.28*i,0.425+0.3028125*i,0.45+0.32625*i,0.475+0.3503125*i,0.5+0.375*i,0.525+0.4003125*i,0.55+0.42625*i,0.575+0.4528125*i,0.6+0.48*i,0.625+0.5078125*i,0.65+0.53625*i,0.675+0.5653125*i,0.7+0.595*i,0.725+0.6253125*i,0.75+0.65625*i,0.775+0.6878125*i,0.8+0.72*i,0.825+0.7528125*i,0.85+0.78625*i,0.875+0.8203125*i,0.9+0.855*i,0.925+0.8903125*i,0.95+0.92625*i,0.975+0.9628125*i,1+1*i,1.025+1.0378125*i,1.05+1.07625*i,1.075+1.1153125*i,1.1+1.155*i,1.125+1.1953125*i,1.15+1.23625*i,1.175+1.2778125*i,1.2+1.32*i,1.225+1.3628125*i,1.25+1.40625*i,1.275+1.4503125*i,1.3+1.495*i,1.325+1.5403125*i,1.35+1.58625*i,1.375+1.6328125*i,1.4+1.68*i,1.425+1.7278125*i,1.45+1.77625*i,1.475+1.8253125*i,1.5+1.875*i,1.525+1.9253125*i,1.55+1.97625*i,1.575+2.0278125*i,1.6+2.08*i,1.625+2.1328125*i,1.65+2.18625*i,1.675+2.2403125*i,1.7+2.295*i,1.725+2.3503125*i,1.75+2.40625*i,1.775+2.4628125*i,1.8+2.52*i,1.825+2.5778125*i,1.85+2.63625*i,1.875+2.6953125*i,1.9+2.755*i,1.925+2.8153125*i,1.95+2.87625*i,1.975+2.9378125*i,2+3*i,2.025+3.0628125*i,2.05+3.12625*i,2.075+3.1903125*i,2.1+3.255*i,2.125+3.3203125*i,2.15+3.38625*i,2.175+3.4528125*i,2.2+3.52*i,2.225+3.5878125*i,2.25+3.65625*i,2.275+3.7253125*i,2.3+3.795*i,2.325+3.8653125*i,2.35+3.93625*i,2.375+4.0078125*i,2.4+4.08*i,2.425+4.1528125*i,2.45+4.22625*i,2.475+4.3003125*i,2.5+4.375*i,2.525+4.4503125*i,2.55+4.52625*i,2.575+4.6028125*i,2.6+4.68*i,2.625+4.7578125*i,2.65+4.83625*i,2.675+4.9153125*i,2.7+4.995*i,2.725+5.0753125*i,2.75+5.15625*i,2.775+5.2378125*i,2.8+5.32*i,2.825+5.4028125*i,2.85+5.48625*i,2.875+5.5703125*i,2.9+5.655*i,2.925+5.7403125*i,2.95+5.82625*i,2.975+5.9128125*i,3+6*i,3.025+6.0878125*i,3.05+6.17625*i,3.075+6.2653125*i,3.1+6.355*i,3.125+6.4453125*i,3.15+6.53625*i,3.175+6.6278125*i,3.2+6.72*i,3.225+6.8128125*i,3.25+6.90625*i,3.275+7.0003125*i,3.3+7.095*i,3.325+7.1903125*i,3.35+7.28625*i,3.375+7.3828125*i,3.4+7.48*i,3.425+7.5778125*i,3.45+7.67625*i,3.475+7.7753125*i,3.5+7.875*i,3.525+7.9753125*i,3.55+8.07625*i,3.575+8.1778125*i,3.6+8.28*i,3.625+8.3828125*i,3.65+8.48625*i,3.675+8.5903125*i,3.7+8.695*i,3.725+8.8003125*i,3.75+8.90625*i,3.775+9.0128125*i,3.8+9.12*i,3.825+9.2278125*i,3.85+9.33625*i,3.875+9.4453125*i,3.9+9.555*i,3.925+9.6653125*i,3.95+9.77625*i,3.975+9.8878125*i,4+10*i,4.025+10.1128125*i,4.05+10.22625*i,4.075+10.3403125*i,4.1+10.455*i,4.125+10.5703125*i,4.15+10.68625*i,4.175+10.8028125*i,4.2+10.92*i,4.225+11.0378125*i,4.25+11.15625*i,4.275+11.2753125*i,4.3+11.395*i,4.325+11.5153125*i,4.35+11.63625*i,4.375+11.7578125*i,4.4+11.88*i,4.425+12.0028125*i,4.45+12.12625*i,4.475+12.2503125*i,4.5+12.375*i,4.525+12.5003125*i,4.55+12.62625*i,4.575+12.7528125*i,4.6+12.88*i,4.625+13.0078125*i,4.65+13.13625*i,4.675+13.2653125*i,4.7+13.395*i,4.725+13.5253125*i,4.75+13.65625*i,4.775+13.7878125*i,4.8+13.92*i,4.825+14.0528125*i,4.85+14.18625*i,4.875+14.3203125*i,4.9+14.455*i,4.925+14.5903125*i,4.95+14.72625*i,4.975+14.8628125*i,5+15*i,5.025+15.1378125*i,5.05+15.27625*i,5.075+15.4153125*i,5.1+15.555*i,5.125+15.6953125*i,5.15+15.83625*i,5.175+15.9778125*i,5.2+16.12*i,5.225+16.2628125*i,5.25+16.40625*i,5.275+16.5503125*i,5.3+16.695*i,5.325+16.8403125*i,5.35+16.98625*i,5.375+17.1328125*i,5.4+17.28*i,5.425+17.4278125*i,5.45+17.57625*i,5.475+17.7253125*i,5.5+17.875*i,5.525+18.0253125*i,5.55+18.17625*i,5.575+18.3278125*i,5.6+18.48*i,5.625+18.6328125*i,5.65+18.78625*i,5.675+18.9403125*i,5.7+19.095*i,5.725+19.2503125*i,5.75+19.40625*i,5.775+19.5628125*i,5.8+19.72*i,5.825+19.8778125*i,5.85+20.03625*i,5.875+20.1953125*i,5.9+20.355*i,5.925+20.5153125*i,5.95+20.67625*i,5.975+20.8378125*i,6+21*i,6.025+21.1628125*i,6.05+21.32625*i,6.075+21.4903125*i,6.1+21.655*i,6.125+21.8203125*i,6.15+21.98625*i,6.175+22.1528125*i,6.2+22.32*i,6.225+22.4878125*i,6.25+22.65625*i,6.275+22.8253125*i,6.3+22.995*i,6.325+23.1653125*i,6.35+23.33625*i,6.375+23.5078125*i,6.4+23.68*i,6.425+23.8528125*i,6.45+24.02625*i,6.475+24.2003125*i,6.5+24.375*i,6.525+24.5503125*i,6.55+24.72625*i,6.575+24.9028125*i,6.6+25.08*i,6.625+25.2578125*i,6.65+25.43625*i,6.675+25.6153125*i,6.7+25.795*i,6.725+25.9753125*i,6.75+26.15625*i,6.775+26.3378125*i,6.8+26.52*i,6.825+26.7028125*i,6.85+26.88625*i,6.875+27.0703125*i,6.9+27.255*i,6.925+27.4403125*i,6.95+27.62625*i,6.975+27.8128125*i,7+28*i,7.025+28.1878125*i,7.05+28.37625*i,7.075+28.5653125*i,7.1+28.755*i,7.125+28.9453125*i,7.15+29.13625*i,7.175+29.3278125*i,7.2+29.52*i,7.225+29.7128125*i,7.25+29.90625*i,7.27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, // fltk 7Fl_Tile 817 1346 120 300 20 [ // fltk N4xcas14Mouse_PositionE 817 1346 120 40 20 [] , // fltk 8Fl_Group 817 1386 120 115 20 [ // fltk 9Fl_Button 817 1386 40 23 20 [] , // fltk 9Fl_Button 857 1386 40 23 20 [] , // fltk 9Fl_Button 897 1386 40 23 20 [] , // fltk 9Fl_Button 817 1409 40 23 20 [] , // fltk 9Fl_Button 857 1409 40 23 20 [] , // fltk 9Fl_Button 897 1409 40 23 20 [] , // fltk 9Fl_Button 817 1432 40 23 20 [] , // fltk 9Fl_Button 857 1432 40 23 20 [] , // fltk 9Fl_Button 897 1432 40 23 20 [] , // fltk 9Fl_Button 817 1455 40 23 20 [] , // fltk 9Fl_Button 857 1455 40 23 20 [] , // fltk 9Fl_Button 897 1455 40 23 20 [] , // fltk 9Fl_Button 817 1478 40 23 20 [] , // fltk 11Fl_Menu_Bar 857 1478 80 23 20 [] ] , // fltk 8Fl_Group 817 1501 120 145 20 [ ] ] ] ] , // fltk 7Fl_Tile 36 1648 901 25 20 [ // fltk N4xcas23Comment_Multiline_InputE 36 1648 901 24 20 Cela semble parfaitement coller...Essayons pour de plus grandes valeurs de n , // fltk N4xcas10Log_OutputE 36 1672 901 1 20 ] , // fltk 7Fl_Tile 36 1675 901 325 20 [ // fltk N4xcas19Multiline_Input_tabE 36 1675 901 24 20 AffSuite(30,0,1,1),plotfunc((1/2)*x^2+(1/2)*x,x=0..30) , // fltk N4xcas10Log_OutputE 36 1699 901 1 20 , // fltk 7Fl_Tile 36 1700 901 300 20 [ // fltk N4xcas7Graph2dE 36 1700 781 300 20 0,30,0,465,[pnt(pnt[0,100663297]),pnt(pnt[1+i,100663297]),pnt(pnt[2+3*i,100663297]),pnt(pnt[3+6*i,100663297]),pnt(pnt[4+10*i,100663297]),pnt(pnt[5+15*i,100663297]),pnt(pnt[6+21*i,100663297]),pnt(pnt[7+28*i,100663297]),pnt(pnt[8+36*i,100663297]),pnt(pnt[9+45*i,100663297]),pnt(pnt[10+55*i,100663297]),pnt(pnt[11+66*i,100663297]),pnt(pnt[12+78*i,100663297]),pnt(pnt[13+91*i,100663297]),pnt(pnt[14+105*i,100663297]),pnt(pnt[15+120*i,100663297]),pnt(pnt[16+136*i,100663297]),pnt(pnt[17+153*i,100663297]),pnt(pnt[18+171*i,100663297]),pnt(pnt[19+190*i,100663297]),pnt(pnt[20+210*i,100663297]),pnt(pnt[21+231*i,100663297]),pnt(pnt[22+253*i,100663297]),pnt(pnt[23+276*i,100663297]),pnt(pnt[24+300*i,100663297]),pnt(pnt[25+325*i,100663297]),pnt(pnt[26+351*i,100663297]),pnt(pnt[27+378*i,100663297]),pnt(pnt[28+406*i,100663297]),pnt(pnt[29+435*i,100663297]),pnt(pnt[30+465*i,100663297]),pnt(pnt[curve(group[pnt[x+(i)*(1/2*x^2+1/2*x),x,0.0,30.0],group[0.0,0.075+0.0403125*i,0.15+0.08625*i,0.225+0.1378125*i,0.3+0.195*i,0.375+0.2578125*i,0.45+0.32625*i,0.525+0.4003125*i,0.6+0.48*i,0.675+0.5653125*i,0.75+0.65625*i,0.825+0.7528125*i,0.9+0.855*i,0.975+0.9628125*i,1.05+1.07625*i,1.125+1.1953125*i,1.2+1.32*i,1.275+1.4503125*i,1.35+1.58625*i,1.425+1.7278125*i,1.5+1.875*i,1.575+2.0278125*i,1.65+2.18625*i,1.725+2.3503125*i,1.8+2.52*i,1.875+2.6953125*i,1.95+2.87625*i,2.025+3.0628125*i,2.1+3.255*i,2.175+3.4528125*i,2.25+3.65625*i,2.325+3.8653125*i,2.4+4.08*i,2.475+4.3003125*i,2.55+4.52625*i,2.625+4.7578125*i,2.7+4.995*i,2.775+5.2378125*i,2.85+5.48625*i,2.925+5.7403125*i,3+6*i,3.075+6.2653125*i,3.15+6.53625*i,3.225+6.8128125*i,3.3+7.095*i,3.375+7.3828125*i,3.45+7.67625*i,3.525+7.9753125*i,3.6+8.28*i,3.675+8.5903125*i,3.75+8.90625*i,3.825+9.2278125*i,3.9+9.555*i,3.975+9.8878125*i,4.05+10.22625*i,4.125+10.5703125*i,4.2+10.92*i,4.275+11.2753125*i,4.35+11.63625*i,4.425+12.0028125*i,4.5+12.375*i,4.575+12.7528125*i,4.65+13.13625*i,4.725+13.5253125*i,4.8+13.92*i,4.875+14.3203125*i,4.95+14.72625*i,5.025+15.1378125*i,5.1+15.555*i,5.175+15.9778125*i,5.25+16.40625*i,5.325+16.8403125*i,5.4+17.28*i,5.475+17.7253125*i,5.55+18.17625*i,5.625+18.6328125*i,5.7+19.095*i,5.775+19.5628125*i,5.85+20.03625*i,5.925+20.5153125*i,6+21*i,6.075+21.4903125*i,6.15+21.98625*i,6.225+22.4878125*i,6.3+22.995*i,6.375+23.5078125*i,6.45+24.02625*i,6.525+24.5503125*i,6.6+25.08*i,6.675+25.6153125*i,6.75+26.15625*i,6.825+26.7028125*i,6.9+27.255*i,6.975+27.8128125*i,7.05+28.37625*i,7.125+28.9453125*i,7.2+29.52*i,7.275+30.1003125*i,7.35+30.68625*i,7.425+31.2778125*i,7.5+31.875*i,7.575+32.4778125*i,7.65+33.08625*i,7.725+33.7003125*i,7.8+34.32*i,7.875+34.9453125*i,7.95+35.57625*i,8.025+36.2128125*i,8.1+36.855*i,8.175+37.5028125*i,8.25+38.15625*i,8.325+38.8153125*i,8.4+39.48*i,8.475+40.1503125*i,8.55+40.82625*i,8.625+41.5078125*i,8.7+42.195*i,8.775+42.8878125*i,8.85+43.58625*i,8.925+44.2903125*i,9+45*i,9.075+45.7153125*i,9.15+46.43625*i,9.225+47.1628125*i,9.3+47.895*i,9.375+48.6328125*i,9.45+49.37625*i,9.525+50.1253125*i,9.6+50.88*i,9.675+51.6403125*i,9.75+52.40625*i,9.825+53.1778125*i,9.9+53.955*i,9.975+54.7378125*i,10.05+55.52625*i,10.125+56.3203125*i,10.2+57.12*i,10.275+57.9253125*i,10.35+58.73625*i,10.425+59.5528125*i,10.5+60.375*i,10.575+61.2028125*i,10.65+62.03625*i,10.725+62.8753125*i,10.8+63.72*i,10.875+64.5703125*i,10.95+65.42625*i,11.025+66.2878125*i,11.1+67.155*i,11.175+68.0278125*i,11.25+68.90625*i,11.325+69.7903125*i,11.4+70.68*i,11.475+71.5753125*i,11.55+72.47625*i,11.625+73.3828125*i,11.7+74.295*i,11.775+75.2128125*i,11.85+76.13625*i,11.925+77.0653125*i,12+78*i,12.075+78.9403125*i,12.15+79.88625*i,12.225+80.8378125*i,12.3+81.795*i,12.375+82.7578125*i,12.45+83.72625*i,12.525+84.7003125*i,12.6+85.68*i,12.675+86.6653125*i,12.75+87.65625*i,12.825+88.6528125*i,12.9+89.655*i,12.975+90.6628125*i,13.05+91.67625*i,13.125+92.6953125*i,13.2+93.72*i,13.275+94.7503125*i,13.35+95.78625*i,13.425+96.8278125*i,13.5+97.875*i,13.575+98.9278125*i,13.65+99.98625*i,13.725+101.0503125*i,13.8+102.12*i,13.875+103.1953125*i,13.95+104.27625*i,14.025+105.3628125*i,14.1+106.455*i,14.175+107.5528125*i,14.25+108.65625*i,14.325+109.7653125*i,14.4+110.88*i,14.475+112.0003125*i,14.55+113.12625*i,14.625+114.2578125*i,14.7+115.395*i,14.775+116.5378125*i,14.85+117.68625*i,14.925+118.8403125*i,15+120*i,15.075+121.1653125*i,15.15+122.33625*i,15.225+123.5128125*i,15.3+124.695*i,15.375+125.8828125*i,15.45+127.07625*i,15.525+128.2753125*i,15.6+129.48*i,15.675+130.6903125*i,15.75+131.90625*i,15.825+133.1278125*i,15.9+134.355*i,15.975+135.5878125*i,16.05+136.82625*i,16.125+138.0703125*i,16.2+139.32*i,16.275+140.5753125*i,16.35+141.83625*i,16.425+143.1028125*i,16.5+144.375*i,16.575+145.6528125*i,16.65+146.93625*i,16.725+148.2253125*i,16.8+149.52*i,16.875+150.8203125*i,16.95+152.12625*i,17.025+153.4378125*i,17.1+154.755*i,17.175+156.0778125*i,17.25+157.40625*i,17.325+158.7403125*i,17.4+160.08*i,17.475+161.4253125*i,17.55+162.77625*i,17.625+164.1328125*i,17.7+165.495*i,17.775+166.8628125*i,17.85+168.23625*i,17.925+169.61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, // fltk 7Fl_Tile 817 1700 120 300 20 [ // fltk N4xcas14Mouse_PositionE 817 1700 120 40 20 [] , // fltk 8Fl_Group 817 1740 120 115 20 [ // fltk 9Fl_Button 817 1740 40 23 20 [] , // fltk 9Fl_Button 857 1740 40 23 20 [] , // fltk 9Fl_Button 897 1740 40 23 20 [] , // fltk 9Fl_Button 817 1763 40 23 20 [] , // fltk 9Fl_Button 857 1763 40 23 20 [] , // fltk 9Fl_Button 897 1763 40 23 20 [] , // fltk 9Fl_Button 817 1786 40 23 20 [] , // fltk 9Fl_Button 857 1786 40 23 20 [] , // fltk 9Fl_Button 897 1786 40 23 20 [] , // fltk 9Fl_Button 817 1809 40 23 20 [] , // fltk 9Fl_Button 857 1809 40 23 20 [] , // fltk 9Fl_Button 897 1809 40 23 20 [] , // fltk 9Fl_Button 817 1832 40 23 20 [] , // fltk 11Fl_Menu_Bar 857 1832 80 23 20 [] ] , // fltk 8Fl_Group 817 1855 120 145 20 [ ] ] ] ] , // fltk 7Fl_Tile 36 2002 901 114 20 [ // fltk N4xcas23Comment_Multiline_InputE 36 2002 901 113 20 C'est convaincant ! On peut alors essayer de prouver ce résultat par récurrence (nous vous laissons le faire...)£Il semble tentant de créer une procédure qui donnerait directement l'expression de u(n) en fonction de n en £donnant comme arguments n,u(0)=u0,a,b.£Le résultat de linsolve étant sous forme d'une liste, A sera L[0], B sera L[1] et C sera L[2] £(Xcas numérote à partir de 0...). On renvoie le résultat en fonction de n. , // fltk N4xcas10Log_OutputE 36 2115 901 1 20 ] , // fltk 7Fl_Tile 36 2118 901 260 20 [ // fltk N4xcas7EditeurE 36 2118 901 173 20 172 , Expression_de_u(n,u0,a,b):={ local L; //ne pas declarer A,B,C,k en local L:=linsolve([(A*k^2+B*k+C=u(k,u0,a,b))$(k=0..2)],[A,B,C]); return L[0]*n^2+L[1]*n+L[2]; }:;, // fltk N4xcas10Log_OutputE 36 2291 901 44 20 // Parsing Expression_de_u£// Warning: A k B C u declared as global variable(s) compiling Expression_de_u£ , // fltk N4xcas8EquationE 36 2335 901 43 16 "Done" ] , // fltk 7Fl_Tile 36 2380 901 68 20 [ // fltk N4xcas19Multiline_Input_tabE 36 2380 901 24 20 Expression_de_u(3,0,1,1) , // fltk N4xcas10Log_OutputE 36 2404 901 1 20 , // fltk N4xcas8EquationE 36 2405 901 43 16 6 ] , // fltk 7Fl_Tile 36 2450 901 102 20 [ // fltk N4xcas19Multiline_Input_tabE 36 2450 901 24 20 Expression_de_u(n,0,1,1) , // fltk N4xcas10Log_OutputE 36 2474 901 1 20 , // fltk N4xcas8EquationE 36 2475 901 77 16 (n^2)/2+n/2 ] , // fltk 7Fl_Tile 36 2554 901 31 20 [ // fltk N4xcas19Multiline_Input_tabE 36 2554 901 30 20 , // fltk N4xcas10Log_OutputE 36 2584 901 1 20 ]