设x1,x2,x3,……,x2007为实数,且满足x1x2x3……x2007=x1-x2x3x4x5……x2007=x1x2-x3x4x5……x2007=……=x1x2x3……x2006-x2007=1,则x2000的值是_________
设x1,x2,x3,……,x2007为实数,且满足x1x2x3……x2007=x1-x2x3x4x5……x2007=x1x2-x3x4x5……x2007=……=x1x2x3……x2006-x2007=1,则x2000的值是_________
设x1,x2,x3,……,x2007为实数,且满足x1x2x3……x2007=x1-x2x3x4x5……x2007=x1x2-x3x4x5……x2007=……=x1x2x3……x2006-x2007=1,则x2000的值是_________
设x1,x2,x3,……,x2007为实数,且满足x1x2x3……x2007=x1-x2x3x4x5……x2007=x1x2-x3x4x5……x2007=……=x1x2x3……x2006-x2007=1,则x2000的值是_________
考虑以下方程组
ab=1
ab=a-b
(a+1)b=a
b=a/(a+1)
a^2/(a+1)=1
a^2-a-1=0
a=(1+-5^0.5)/2
b
=(1+-5^0.5)/(3+-5^0.5)
=(-2+-2*5^0.5)/(9-5)
=(-1+-5^0.5)/2
---------------------------------------------
因
x1x2...x2007=1
x1-x2x3...x2007=1
故
x1=(1+-5^0.5)/2
x2x3...x2007=(-1+-5^0.5)/2
同理
x1x2=(1+-5^0.5)/2(不一定与上组解同号)
x3x4...x2007=(-1+-5^0.5)/2
x1x2x3=(1+-5^0.5)/2(不一定与上组解同号)
x4x5...x2007=(-1+-5^0.5)/2
...
x1x2...x2006=(1+-5^0.5)/2(不一定与上组解同号)
x2007=(-1+-5^0.5)/2
x2=x1x2/x1=(1+-5^0.5)/(1+-5^0.5)=1
或x2=x1x2/x1=(1-+5^0.5)/(1+-5^0.5)=(-3+-5^0.5)/2
因此
x1=(1+-5^0.5)/2
x2=1,(-3+-5^0.5)/2(与x1中的根号5同号)
同理x3=1,(-3+或-5^0.5)/2(与上方最相邻的不为1的解中的根号5反号)
x4=1,(-3+或-5^0.5)/2(与上方最相邻的不为1的解中的根号5反号)
...
x2006=1,(-3+或-5^0.5)/2(与上方最相邻的不为1的解中的根号5反号)
x2007=(1+或-5^0.5)/2(若x2...x2006中有偶数个不为1,则x1与x2007的根号5同号,否则为反号)
共2^2006组解
x2000有三个可能值:1,(-3+5^0.5)/2,(-3-5^0.5)/2
例:
x1=(1+5^0.5)/2
x2=(-3+5^0.5)/2
x3=x4=...=x2006=1
x2007=(-1-5^0.5)/2
为一组解
x1=(1+5^0.5)/2
x2=x3=...=x1999=1
x2000=(-3+5^0.5)/2
x2001=...=x2006=1
x2007=(-1-5^0.5)/2
也是一组解
x1=(1+5^0.5)/2
x2=(-3+5^0.5)/2
x3=x4=...=x1999=1
x2000=(-3-5^0.5)/2
x2001=...=x2006=1
x2007=(-1+5^0.5)/2
也是一组解