作业帮排序不等式相关例题
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排序不等式相关例题
排序不等式相关例题
排序不等式相关例题
1.设a1,a2,a3为正数,求证：(a1*a2)/a3+(a2*a3)/a1+(a3*a1)/a2≥a1+a2+a3
不妨设
a1≥a2≥a3
则a1a2≥a1a3≥a2a3
(a1*a2)/a3+(a2*a3)/a1+(a3*a1)/a2≥a1+a2+a3
即证(a1a2)^2+(a2a3)^2+(a1a3^2)≥(a1+a2+a3)a1a2a3
(a1a2)^2+(a2a3)^2+(a1a3)^2≥a1^2a2a3+a1a2^2a3+a1a2a3^2
同序和≥乱序和
(a1a2)^2+(a2a3)^2+(a1a3)^2≥a1a2*a1a3+a1a2*a2a3+a1a3*a2a3
即(a1a2)^2+(a2a3)^2+(a1a3)^2≥a1^2a2a3+a1a2^2a3+a1a2a3^2
则原不等式得证
2.设x>0,求证1+x+x^2+x^3+.+x^(2n)>=(2n+1)*x^n
给你两种证法:
1.用排序不等式:
1+x+x^2+...+x^2n
=x^0*x^0+x^(1/2)*x^(1/2)+x^1*x^1+...x^(n-2/2)*x(n-2/2)+x^(n-1/2)*x^(n-1/2)+x^n*x^n(顺序和)
≥x^0*x^n+x^(1/2)*(x^(n-1/2)+x^1*x^(n-2/2)+...+x^(n-1/2)*x^(1/2)+x^n*x^0(乱序和)
=x^n+x^n+x^n+...+x^n
=(2n+1)x^n
等号成立当且仅当x=1
2.用基本不等式,算术平均≥几何平均
1+x+x^2+...+x^n
≥(2n+1)(1*x*x^2*..*x^2n)(1/(2n+1))
=(2n+1)(x^(2n+1)*n)^(1/(2n+1))
=(2n+1)x^n
等号成立当且仅当x=1