Ca semn de bunavointa ma leg eu de primul mesaj codificat.
Incerc sa il transcriu in limba romana, sa il paginez incat cineva sa il mai suporte pe ecran si eventual sa dau o indicatie ca sa putem rezolva ceva impreuna.
(1) Bun.
Sa incercam sa vedem cam care sunt primii termeni ai sirului.
Deoarece stiu sa programez, voi tipari ceva ca sa vedem si noi termenii aia cum arata mai exact.
Cod care ne arata care sunt primii termeni ai sirului...
(21:32) gp > x=1
%3 = 1
(21:32) gp > for( n=2,20, x=1-1/(4*x); print( "x(",n,") = ", x , " ~ " , x+0.0 ) )
x(2) = 3/4 ~ 0.7500000000000000000000000000
x(3) = 2/3 ~ 0.6666666666666666666666666667
x(4) = 5/8 ~ 0.6250000000000000000000000000
x(5) = 3/5 ~ 0.6000000000000000000000000000
x(6) = 7/12 ~ 0.5833333333333333333333333333
x(7) = 4/7 ~ 0.5714285714285714285714285714
x(8) = 9/16 ~ 0.5625000000000000000000000000
x(9) = 5/9 ~ 0.5555555555555555555555555556
x(10) = 11/20 ~ 0.5500000000000000000000000000
x(11) = 6/11 ~ 0.5454545454545454545454545455
x(12) = 13/24 ~ 0.5416666666666666666666666667
x(13) = 7/13 ~ 0.5384615384615384615384615385
x(14) = 15/28 ~ 0.5357142857142857142857142857
x(15) = 8/15 ~ 0.5333333333333333333333333333
x(16) = 17/32 ~ 0.5312500000000000000000000000
x(17) = 9/17 ~ 0.5294117647058823529411764706
x(18) = 19/36 ~ 0.5277777777777777777777777778
x(19) = 10/19 ~ 0.5263157894736842105263157895
x(20) = 21/40 ~ 0.5250000000000000000000000000
Cer acum parerea generala... vem cumva o formula pentru termenul general al sirului? Daa, care? Si cum stam cu convergenta.
(Eu nu vreau sa oblig pe nimeni sa programeze, dar de ce ma obliga soarta...?)
(2) Daca masina de calculat a ajutat deja, inseamna ca va mai putea ajuta poate.
Cod care ne arata care sunt primii termeni ai sirurilor...
(21:39) gp > x=1;y=2;
(21:39) gp > for( n=1,20, x = (x+y)/4; y = (3*x+2*y)/5 ; print( "x(",n,") = ", x , " ~ " , x+0.0 , "\ny(", n, ") = ", y , " ~ " , y+0.0 , "\n" ); )
x(1) = 3/4 ~ 0.7500000000000000000000000000
y(1) = 5/4 ~ 1.250000000000000000000000000
x(2) = 1/2 ~ 0.5000000000000000000000000000
y(2) = 4/5 ~ 0.8000000000000000000000000000
x(3) = 13/40 ~ 0.3250000000000000000000000000
y(3) = 103/200 ~ 0.5150000000000000000000000000
x(4) = 21/100 ~ 0.2100000000000000000000000000
y(4) = 83/250 ~ 0.3320000000000000000000000000
x(5) = 271/2000 ~ 0.1355000000000000000000000000
y(5) = 2141/10000 ~ 0.2141000000000000000000000000
x(6) = 437/5000 ~ 0.08740000000000000000000000000
y(6) = 863/6250 ~ 0.1380800000000000000000000000
x(7) = 5637/100000 ~ 0.05637000000000000000000000000
y(7) = 44527/500000 ~ 0.08905400000000000000000000000
x(8) = 9089/250000 ~ 0.03635600000000000000000000000
y(8) = 35897/625000 ~ 0.05743520000000000000000000000
x(9) = 117239/5000000 ~ 0.02344780000000000000000000000
y(9) = 926069/25000000 ~ 0.03704276000000000000000000000
x(10) = 189033/12500000 ~ 0.01512264000000000000000000000
y(10) = 93323/3906250 ~ 0.02389068800000000000000000000
x(11) = 2438333/250000000 ~ 0.009753332000000000000000000000
y(11) = 19260343/1250000000 ~ 0.01540827440000000000000000000
x(12) = 3931501/625000000 ~ 0.006290401600000000000000000000
y(12) = 15527423/1562500000 ~ 0.009937550720000000000000000000
x(13) = 50712351/12500000000 ~ 0.004056988080000000000000000000
y(13) = 400575821/62500000000 ~ 0.006409213136000000000000000000
x(14) = 81767197/31250000000 ~ 0.002616550304000000000000000000
y(14) = 161469353/39062500000 ~ 0.004133615436800000000000000000
x(15) = 1054713397/625000000000 ~ 0.001687541435200000000000000000
y(15) = 8331159487/3125000000000 ~ 0.002665971035840000000000000000
x(16) = 1700590809/1562500000000 ~ 0.001088378117760000000000000000
y(16) = 6716465957/3906250000000 ~ 0.001719415284992000000000000000
x(17) = 21935885959/31250000000000 ~ 0.0007019483506880000000000000000
y(17) = 173271113189/156250000000000 ~ 0.001108935124409600000000000000
x(18) = 35368817873/78125000000000 ~ 0.0004527208687744000000000000000
y(18) = 34922195851/48828125000000 ~ 0.0007152065710284800000000000000
x(19) = 456221656173/1562500000000000 ~ 0.0002919818599507200000000000000
y(19) = 3603685502983/7812500000000000 ~ 0.0004612717443818240000000000000
x(20) = 735599222981/3906250000000000 ~ 0.0001883134010831360000000000000
y(20) = 2905241585963/9765625000000000 ~ 0.0002974967384026112000000000000
Sper ca e relativ clar unde se duc lucrurile.
Din pacate acum ne incepe afacerea! Trebuie sa izolam noi cu mana proprie ceva proprietati despre sirurile de mai sus care sa ne permita sa concludem convergenta la zero. Probabil ca vedem monotonia ambelor siruri. Cum o demonstram (inductiv)? mai exact: Ce afirmatie putem etabla si demonstra inductiv?
Desigur ca putem calcula si termenul general explicit sau daca suntem pe a XI-a mai dezghetati ne putem lega de matricea si de definitia recursiva...
si putem fie sa calculam explicit puterile matricii A, fie sa ne multumim cu calculul valorilor ei proprii care sunt in modul strict mai mici decat unu.
Deoarece aici e un forum ma opresc aici, dar discutii sunt bine venite.
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