A.M., G.M., H.M. and Their Relations

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A.M., G.M., H.M. and Their Relations: Overview

This Topic covers sub-topics such as Relation between A.M., G.M. and H.M., AM-GM-HM Inequality, Relationship between Arithmetic Mean and Geometric Mean of Two Numbers and, Arithmetic Mean of mth Power Inequality

Important Questions on A.M., G.M., H.M. and Their Relations

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If a1,a2,.....an are positive real numbers whose product is a fixed number e, the minimum value of a1+a2+a3+.......+an-1+2an is

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If a1, a2, .......an are positive real numbers whose product is a fixed number e, the minimum value of a1+a2+ .......+2an is

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If G is the geometric mean of numbers 1, 2, 22, 23,, 2n-1, then what is the value of 1+2log2G?

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Let a,b,c and d be positive real numbers such that a+b+c+d=11. If the maximum value of a5b3c2d is 3750β, then the value of β is

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Minimum value of a+b+c+d1a+1b+1c+1d is 

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If x,y,z>0 and x+y+z=1 then xyz1-x1-y1-z is necessarily.

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If xyz=1-x1-y1-z where, 0x,y,z1, then the minimum value of x1-z+y1-x+z1-y is:

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The minimum value of x2+x+1y2+y+1xy where x,y >0 is

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For a distinct complex numbers z1,z2,z3,znn3, the min value of z2-z12+z3-z22++zn-zn-12zn-z12 

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If x,y,z are positive integers, then x+yy+zz+x is

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If a,b,c are non-zero real numbers, then the minimum value of the expression a8+4a4+1b8+3b4+1c2+2c+2a4b4 equals

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If x+y=15, and x2y3 is maximum, then

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The sum of square of any real positive quantities and its reciprocal is never less

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If e=li^+mj^+nk^ is a unit vector and l=13, then maximum value of lmn is

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Find the number of real ordered pair(s) x,y for which: 16x2+y+16x+y2=1

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If A.M of two numbers is 11 and their G.M. is 230, find the two numbers.
 

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If A.M of two numbers be twice their G.M then the numbers are in the ratio.

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For the two data points 4, 5, find the relationship between AM, GM, HM, prove that, AM > GM > HM.

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For the two data points 10, 12, find the relationship between AM, GM, HM, prove that, AM > GM > HM.

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Find the value of geometric mean GM, if the arithmetic mean AM is 7, and harmonic mean HM is 487.