函数+数列题已知函数f(x)=x²+x-1,α,β为方程以f(x)=0的两个根(α>β),f'(x)是f(x)的导数,设A1=1,A(n+1)=An-[f(An)/f'(An)](n=1,2,3…)1)求α,β的值.2)已知对任意的正整数n有An>α,记Bn=ln[(An-β)/(An-α)](n=1,2,3...),

Question

函数+数列题
已知函数f(x)=x²+x-1,α,β为方程以f(x)=0的两个根(α>β),f'(x)是f(x)的导数,设A1=1,A(n+1)=An-[f(An)/f'(An)](n=1,2,3…) 1)求α,β的值.2)已知对任意的正整数n有An>α,记Bn=ln[(An-β)/(An-α)](n=1,2,3...),求数列{Bn}的前n项和Sn

Answer 1

1.α=（-1+√5）/2,β=（-1-√5）/2.
f'(x)=2x+1,
A=An-[An^2+An-1]/[2An+1]
=(An^2+1)/(2An+1),
[A-β]/[A-α]
=[An^2+1+(1+√5）（An+1/2)]/[An^2+1+(1-√5）（An+1/2)]
=[An^2+(1+√5）An+(3+√5）/2]/[An^2+(1-√5）An+(3-√5）/2]
=[(An-β)/(An-α)]^2,
∴B=2Bn,
∴数列{Bn}是以B1=ln[(1-β)/(1-α)]=2ln[(3+√5）/2]为首项,2为公比的等比数列,
∴Sn=2ln[(3+√5）/2]*(2^n-1).
αβ