Question

有一题目,想请教一下?(高中)已知f(x)=a1x1+a2x2+a3x3+```+anxn,且f(1)=n2,求f(1/

Answer 1

f(1)=a1+a2+a3+.+an=n^2 (1)

所以 a1+a2+a3+.an+1=(n+1)^2 (2)

(2)-(1) 得 an+1-an=2n+1 所以 an-a1=n^2-1

因为 a1+a2+a3+.+an=n^2 所以n=1时 a1=1 得 an=n^2

f(1/2)=a1*(1/2)+.an*(1/2)^n (3)

(1/2)*f(1/2)=a1*(1/4)+.an(1/2)^(n+1) (4)

(4)-(3) 得 (1/2)*f(1/2)=a1*(1/2)+(a2-a1)*(1/4)+.(an-an-1)*(1/2)^n-an*(1/2)^(n+1)

(1/2)*f(1/2)=a1*(1/2)+(2n+1)*(1/4)+.(2n+1)*(1/2)^n-an*(1/2)^(n+1)

(1/2)*f(1/2)=1/2+3/2-3*(1/2)n+n^2*(1/2)^n+1

所以 f(1/2)=(n^2-6)*(1/2)^n+4

应该是这样做的吧 关键是求等差和等比数列 如果做的不对 就不要看了