Question

If A,G and H are A.M, G.M, H.M OF 2 POSITIVE NUMBERS X AND Y RESPECTIVELY THEN PROVE THAT:
G^2= A.H

Answer 1

Answer:

See below.

Step-by-step explanation:

A: Arithmetic Mean

G: Geometric Mean

H: Harmonic Mean

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Derivation of AM × HM = $GM^2$

Arithmetic Progression

x, AM, y   →   arithmetic progression

Taking the common difference of arithmetic progression,

AM − x = y − AM

x + y = 2 AM   →   Equation (1)

Geometric Progression

x, GM, y   →   geometric progression

The common ratio of this geometric progression is

$\frac{GM}{x}=\frac{y}{GM}$

$xy=GM^2$   →   Equation (2)

Harmonic Progression

x, HM, y   →   harmonic progression

$\frac{1}{x},\frac{1}{HM},\frac{1}{y}$   →   the reciprocal of each term will form an arithmetic progression

The common difference is

$\frac{1}{HM} -\frac{1}{x}=\frac{1}{y}-\frac{1}{HM}$

$\frac{2}{HM} =\frac{1}{x}+\frac{1}{y}$

$\frac{2}{HM} =\frac{X+y}{xy}$  →   Equation (3)

Substitute x + y = 2AM from Equation (1) and xy = $GM^2$ from Equation (2) to Equation (3)

$\frac{2}{HM} =\frac{2AM}{GM^2}$

$GM^2=$AM x HM →   PROVED!

Answer 2

Answer:

I am Don maths is very bad subject

Step-by-step explanation:

So sorry I am not interested in maths