数列{Xn}各项均为正,满足x1^2+x2^2+...+Xn^2=2*n^2+2*n .

题目

数列{Xn}各项均为正,满足x1^2+x2^2+...+Xn^2=2*n^2+2*n .
(1) 求Xn.
(2) 已知1/(x1+x2)+1/(x2+x3)+...+1/(Xn+Xn+1）=3,求n.
(3) 证明X1*X2+X2*X3+...+Xn*Xn+1

答案

(1)x1^2+x2^2+...+Xn^2=2*n^2+2*n
x1^2+x2^2+...+X(n-1)^2=2*(n-1)^2+2*(n-1)
Xn^2=4n Xn=2√n
(2)1/(x1+x2)+1/(x2+x3)+...+1/(Xn+Xn+1）=1/(2√1+2√2)+1/(2√2+2√3)+...+1/(2√n+2√(n+1)）={(√2-√1)+(√3-√2)+...+(√(n+1)-√n)}*1/2=
1/2*(√(n+1)-1)=3 n=48
(3)X1*X2+X2*X3+...+Xn*Xn+1=2(2√1√2+2√2√3+...+2√n√(n+1))