若a,b,x,y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

题目

若a,b,x,y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

答案

F（n）=ax^n+by^n
x,y是tt＝pt+q两根
F（n)=pF(n-1)+qF(n-2)
F(3)=pF(2)+qF(1)
F(4)=pF(3)+qF(2)
p=-14
q=38
F(5)=20