Question

已知ax+by=7,ax^2+by^2=49,ax^3+by^3=133,ax^4+by^4=406,试求1995（x+y)+6xy-2分之17（a+b)

Answer 1

已知ax+by=7,ax^2+by^2=49,ax^3+by^3=133,ax^4+by^4=406,求1995(x+y)+6xy-17/2*(a+b)的值.
在ax^2+by^2=49的两边同乘以(x+y),得
(x+y)(ax^2+by^2)=49(x+y)
展开即得：
ax^3+bxy^2+ax^2y+by^3=49(x+y)
ax^3+by^3+xy(ax+by)=49(x+y)
133+7xy=49(x+y)
化简得：
19+xy=7(x+y)···········①
在ax^3+bx^3=133的两边同乘以(x+y),得
(x+y)(ax^3+by^3)=133(x+y)
ax^4+bxy^3+ax^3y+by^4=133(x+y)
ax^4+by^4+xy(ax^2+by^2)=133(x+y)
406+49xy=133(x+y)
58+7xy=19(x+y)··········②
为方便求解,设x+y=m,xy=n,①、②联立方程组,
19+n=7m
58+7n=19m
解之得：m=5/2,n=-3/2,
即：x+y=5/2,xy=-3/2,由x+y=5/2得：y=5/2-x,代入xy=-3/2,得
x(5/2-x)=-3/2
x^2-5/2x-3/2=0
(x+1/2)(x-3)=0
x=-1/2和3,相应的y=3和-1/2；
为方便起见,令x>y,则有
x=3,y=-1/2,代入ax+by=7,ax^2+by^2=49,得
3a-1/2b=7
9a+1/4b=49
解这个方程组,得：a=5,b=16
所以
1995(x+y)+6xy-8.5×(a+b)
=1995m+6n-8.5×(a+b)
=1995×5/2+6×(-3/2)-8.5×(5+16)
=4800