已知a+b=1,a 2 +b 2 =2,求a 7 +b 7 的值.
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∵a+b=1,
∴(a+b) 2 =1,
∴a 2 +b 2 +2ab=1.
∵a 2 +b 2 =2,
∴2+2ab=1,
∴ab=-0.5,
∴(a 2 +b 2 ) 2 =4,
则有a 4 +b 4 +2(ab) 2 =4,
∴a 4 +b 4 =3.5,
平方得:(a 4 +b 4 ) 2 =12.25,
a 8 +b 8 +2(ab) 4 =12.25,
∴a 8 +b 8 =12.125,
∵a 6 +b 6 =(a 2 +b 2 )[a 4 +b 4 -(ab) 2 ]
=2×(3.5-0.25)
=6.5,
∴a 7 +b 7 =(a 7 +b 7 )(a+b)
=a 8 +b 8 +ab 7 +ba 7
=(a 8 +b 8 )+ab(a 6 +b 6 )
=12.125+(-0.5)×6.5
=8.875.
∴(a+b) 2 =1,
∴a 2 +b 2 +2ab=1.
∵a 2 +b 2 =2,
∴2+2ab=1,
∴ab=-0.5,
∴(a 2 +b 2 ) 2 =4,
则有a 4 +b 4 +2(ab) 2 =4,
∴a 4 +b 4 =3.5,
平方得:(a 4 +b 4 ) 2 =12.25,
a 8 +b 8 +2(ab) 4 =12.25,
∴a 8 +b 8 =12.125,
∵a 6 +b 6 =(a 2 +b 2 )[a 4 +b 4 -(ab) 2 ]
=2×(3.5-0.25)
=6.5,
∴a 7 +b 7 =(a 7 +b 7 )(a+b)
=a 8 +b 8 +ab 7 +ba 7
=(a 8 +b 8 )+ab(a 6 +b 6 )
=12.125+(-0.5)×6.5
=8.875.
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