若abxy满足ax+by=3,ax^2+by^2=7,ax^3+by^3=16,ax^4+by^4=42,求ax^5+by^5的值

Question

若abxy满足ax+by=3,ax^2+by^2=7,ax^3+by^3=16,ax^4+by^4=42,求ax^5+by^5的值

Answer 1

∵ (ax^2+by^2)·(x+y)=(ax^3+by^3)+xy(ax+by),
∴ 7(x+y)=16+3xy ………………①
∵ (ax^3+by^3)·(x+y)=(ax^4+by^4)+xy(ax^2+by^2),
∴ 16(x+y)=42+7xy ………………②∵ (ax^4+by^4)·(x+y)=(ax^5+by^5)+xy(ax^3+by^3),
∴ 42(x+y)=(ax^5+by^5)+16xy ……③
由①和②可得,xy=-38；(x+y)=-14
代入③,可得：ax^5+by^5=20