已知 x+y+z=1 ，x^2+y^2+z^2=2 ，x^3+y^3+z^3=3 ，求 x^5+y^5+z^5

图老师3

Solve If You Are A Genius (Only 1% Can)

luyuanhong

图老师

Clear["Global`*"];(*Clear all \
variables*)(*求解出方程的根*)(*解析解求解累机器，所以用数值解，虽然数值，但是求解很快*)aa =
NSolve[x + y + z == 1 && x^2 + y^2 + z^2 == 2 &&
   x^3 + y^3 + z^3 == 3, {x, y, z}, 200];
bb = {};(*用来保存求解结果的变量*)Do[
cc = Union@RootApproximant[x^k + y^k + z^k /. aa];(*用数值来得到近似值*)
bb = Append[bb, Flatten@{k, cc}],(*结果的集合*){k, 5, 100}];
Print[bb]

图老师

{5,6},{6,103/12},{7,221/18},{8,1265/72},{9,905/36},{10,15539/432},{11,11117/216},{12,21209/288},{13,136561/1296},{14,260531/1728},{15,559171/2592},{16,9601075/31104},{17,6868847/15552},{18,39313147/62208},{19,84376799/93312},{20,482922025/373248},{21,38388265/20736},{22,5932205891/2239488},{23,1414682371/373248},{24,24290371883/4478976},{25,52133802109/6718464},{26,298382512537/26873856},{27,213470244319/13436928},{28,3665325677875/161243136},{29,2622264828163/80621568},{30,5002755322601/107495424},{31,32211856275131/483729408},{32,252896134865/2654208},{33,131896630417325/967458816},{34,2264690837058179/11609505792},{35,1620215950939357/5804752896},{36,9273141162484907/23219011584},{37,19902705015074521/34828517376},{38,113911107351510233/139314069504},{39,27164943090690697/23219011584},{40,1399282093379693875/835884417024},{41,333693696060606109/139314069504},{42,5729585763195255547/1671768834048},{43,12297262956028185239/2507653251072},{44,70382133466425780905/10030613004288},{45,50353122570838703465/5015306502144},{46,864572923073454525251/60183678025728},{47,618536895982456263473/30091839012864},{48,393348110809626043667/13374150672384},{49,7598096645414047642549/180551034077184},{50,14495644385815306267699/240734712102912},{51,31111629722780324240599/361102068154368},{52,534192742727357837866675/4333224817852416},{53,382174727110742851056683/2166612408926208},{54,2187338170061450539055803/8666449635704832},{55,4694627807788083280687091/12999674453557248},{56,26869224649779656144142505/51998697814228992},{57,6407636405902878874095415/8666449635704832},{58,330061095792994941022610819/311992186885373952},{59,26237098652723689827803933/17332899271409664},{60,1351488284161882321312697387/623984373770747904},{61,2900664637762860577831057681/935976560656121856},{62,16601658955035142109063264473/3743906242624487424},{63,11877238257361372793019038611/1871953121312243712},{64,203934494504493342154742487475/22463437455746924544},{65,145899791501848360510959991807/11231718727873462272},{66,278347537847110758456886203049/14975624970497949696},{67,1792230542069790165666239132879/67390312367240773632},{68,3419216391636850310273590775395/89853749822987698176},{69,7338576436295731768761954600545/134780624734481547264},{70,126004786928559594680270728716611/1617367496813778567168},{71,90146947360622268111831614323033/808683748406889283584},{72,515946882116231594202645004410923/3234734993627557134336},{73,1107363558720493992979597713943789/4852102490441335701504},{74,6337882670673159826973652326300057/19408409961765342806016},{75,55978744877965304982317856534559/119804999763983597568},{76,77854442268283731895652304386266675/116450459770592056836096},{77,18566332491608004990672072498060401/19408409961765342806016},{78,318787545508040550597988159880188603/232900919541184113672192},{79,684205531820674736760857044495959851/349351379311776170508288},{80,3915980753706116658725457527601130025/1397405517247104682033152},{81,2801589681427753341422579888236619165/698702758623552341016576},{82,48103840565534089020290759419626659459/8384433103482628092198912},{83,34414679703903362863775848138323752237/4192216551741314046099456},{84,65656306035619517947028157175289381689/5589622068988418728132608},{85,422749336554764963863099631413586804041/25153299310447884276596736},{86,268840313261694329825639481420582962257/11179244137976837456265216},{87,1731015205285818724999019533569144206731/50306598620895768553193472},{88,29721868267714631460732898926839817677875/603679183450749222638321664},{89,21263761159392396389578885730519964789287/301839591725374611319160832},{90,121700973726427491919044519983196253661947/1207358366901498445276643328},{91,261203678201675742777669886629774183699719/1811037550352247667914964992},{92,1494972678623555032426128649390765354941865/7244150201408990671659859968},{93,356513501458889001650085490438918329333875/1207358366901498445276643328},{94,18364218801197372127792143150492452655643971/43464901208453944029959159808},{95,4379405751013488198319968437963957989765931/7244150201408990671659859968},{96,75195249830865898527432790291326968484841643/86929802416907888059918319616},{97,161389635906018927639924972953912300962977829/130394703625361832089877479424},{98,923697162128835264521837883945197605183627417/521578814501447328359509917696},{99,660835841885869509164560310072174587797212359/260789407250723664179754958848},{100,11346680132641015614099937264517337072969341875/3129472887008683970157059506176}}

王守恩

图老师 发表于 2019-12-6 14:34
{5,6},{6,103/12},{7,221/18},{8,1265/72},{9,905/36},{10,15539/432},{11,11117/216},{12,21209/288},{13, ...

    ”爬楼梯“问题
x^1+y^1+z^1=1
x^2+y^2+z^2=2
x^3+y^3+z^3=3
............
x^n+y^n+z^n=a(n)
LinearRecurrence[{ 1, 1/2 , 1/6 }, { 1, 2, 3 }, n]

     ”爬楼梯“问题
x^1+y^1+z^1=1
x^2+y^2+z^2=2
x^3+y^3+z^3=5
............
x^n+y^n+z^n=a(n)
LinearRecurrence[{ 1, 1/2 , 5/6 }, { 1, 2, 5 }, n]

     一般地，”爬楼梯“问题
x^1+y^1+z^1=a
x^2+y^2+z^2=b
x^3+y^3+z^3=c
............
x^n+y^n+z^n=a(n)
a(n)=(x+y+z)a(n-1)-(xy+yz+zx)a(n-2)+xyza(n-3),
或：a(n)=a(n-1)+1/2a(n-2)+1/6a(n-3).
LinearRecurrence[{a, (b-a^2)/2 , (2c-3ab+a^3)/6}, {a, b, c}, n]

     特别，”爬楼梯“问题
x^1+y^1+z^1=1
x^2+y^2+z^2=3
x^3+y^3+z^3=4
............
x^n+y^n+z^n=a(n)
LinearRecurrence[{1, 1, 0}, {1, 3, 4}, n]
a(n)=1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843,
1364, 2207, 3571, 5778, 9349, 15127, 24476, 39603, 64079,
103682, 167761, 271443, 439204, 710647, 1149851, 1860498,
3010349, 4870847, 7881196, 12752043, 20633239, ................

      说明
LinearRecurrence=线性递推函数

jiangmingdar

看到一個影片上面談到
newton identity
ke(k)=e(k-1)p(1)-e(k-2)p(2)+e(k-3)p(3)-e(k-4)p(4)+....
其中
e(0)=1,e(1)=x+y+z,e(2)=xy+xz+yz,e(3)=xyz,e(4)=0,e(5)=0
p(1)=x+y+z,p(2)=x^2+y^2+z^2,p(3)=x^3+y^3+z^3,p(4)=x^4+y^4+z^4,....
帶入
k=1,k=2,k=3,k=4,....
就可以順利解出答案

jiangmingdar

王守恩

王守恩 发表于 2019-12-6 17:07
”爬楼梯“问题
x^1+y^1+z^1=1
x^2+y^2+z^2=2

这两道题还是有不一样的。
(x+1)(y+1)(z+1)=20,
(x+2)(y+2)(z+2)=21,
(x+3)(y+3)(z+3)=2021,
求 (x+10)(y+10)(z+10)