Question

4 If one geometric mean G, and two arithmetic means p and q, be inserted betwen two given quantities, then prove that G2 = (2p-q) (2q-p)​

Answer 1

$\large\underline{\sf{Solution-}}$

Let assume that

• a and b are two positive real numbers.

Such that,

One geometric mean G, and two arithmetic means p and q, be inserted betwen two given quantities a and b.

Now,

Geometric mean G is inserted between a and b.

$\rm \implies\:a,G,b \: are \: in \: GP$

We know, if numbers are in GP, they have same common ratio.

$\rm \implies\:\dfrac{G}{a} = \dfrac{b}{G}$

$\bf\implies \: {G}^{2} = ab - - - (1)$

Further, given that

Two Arithmetic mean p and q are inserted between two numbers a and b.

$\rm \implies\:a,p,q,b \: are \: in \: AP$

We know, if numbers are in AP, they have same common difference

So,

$\rm \implies\:p - a = q - p = b - q$

$\rm \implies\:p - a = q - p \: \: and \: \: q - p= b - q$

$\rm \implies\:a = 2p - q \: \: and \: \: b = 2q - p$

On substituting these values of a and b in equation (1), we get

$\bf\implies \: {G}^{2} = (2p - q)(2q - p)$

Hence, Proved

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MORE TO KNOW

Let us consider two positive real numbers a and b, then

1. Arithmetic mean (A) between a and b is

$\boxed{\tt{ A = \frac{a + b}{a}}}$

2. Geometric mean (G) between a and b is

$\boxed{\tt{ G = \sqrt{ab}}}$

3. Harmonic mean (H) between a and b is

$\boxed{\tt{ H = \frac{2ab}{a + b}}}$

4. Relationship between Arithmetic mean, Geometric mean and Harmonic mean

$\boxed{\tt{ \: A \: \geqslant \: G \: \geqslant \: H \: }}$

$\boxed{\tt{ \: {G}^{2} \: = \: A \: H \: }}$