已知数列An是等差数列,公差d不等于0,An不等于0,(n属于正整数)A(k)X的平方+2A(k+1)X+A(k+2)=0,(k属_数学解答

已知数列An是等差数列,公差d不等于0,An不等于0,(n属于正整数)
A(k)X的平方+2A(k+1)X+A(k+2)=0,(k属于正整数).
(1)求证:当k取不同的正整数时,方程都有实数根.
(2)若方程不同的根依次为X1,X2,X3.Xn.求证:1/X1+1,1/X2+1,1/X3+1,.,1/Xn+1,.是等差数列.

人气：136 ℃　时间：2020-04-11 00:52:35

解答

【解】
（1）方程A(k)(X^2)+2A(k+1)X+A(k+2)=0,则其
Δ=4[ A(k+1)^2-A(k)*A(k+2)]
=4[ [A(k)+d]^2-A(k)*[A(k)+2d] ]
=4d^2>0;
所以有实数解；
（2）设A(k)(X^2)+2A(k+1)X+A(k+2)=0的根为X(k),X(k+1);则：
X(k)+X(k+1)=-2A(k+1)/A(k)=-2[(A(k)+d)/A(k)]=-2[1+d/A(k)];
X(k)*X(k+1)=A(k+2)/A(k)=1+2d/A(k);
所以：
1/[X(k+1)+1]-1/[X(k)+1]
=[ X(k)-X(k+1) ]/[X(k+1)*X(k)+( X(k+1)+X(k) )+1]
=[(X(k)+X(k+1))^2-4X(k)*X(k+1)]^(1/2)/[X(k+1)*X(k)+( X(k+1)+X(k) )+1]
=(4d^2/A(k)^2)^(1/2)/(-d/A(k))
=|2d/A(k)|/[-d/A(k)]
= 2
所以{1/(X(k)+1)}成等差数列