Question

Let α be the arithmetic mean and β , ω be the geometric means between any two positive numbers, then find the value of ( β³ + ω³) / αβω.

Answer 1

Taking the numbers as a and b and note that  I will convert everything at last to a and b . Then only the answer will come .

Arithmetic mean = sum of terms / number of terms .

⇒ α = ( a + b ) / 2

Now G.Ms lie between the 2 positive numbers . I have read the question wrong earlier . I read that 2 geometric means of a and b are β , ω .

G.Ms lie between a and b . So let a be the first term of the resultant G.P .

The G.P will be something like this one :

a , β , ω , b .

Let the common ratio be r .

b is the 4 th term and hence b = a r³ ⇒ b/a = r³ .

Taking cube root both sides :

∛( b/a ) = r [ I am substituting r so that later I can get only a and b ]

β = ar

⇒ β = a × ∛( b/a )

Similarly we have :

ω = ar²

⇒ ω = a × ∛ ( b²/a² )

Now that we are left with nothing but calculations , I think it would be fair enough to use latex ...

$\dfrac{\beta^3+\omega^3}{\alpha\beta\omega}\\\\\implies \dfrac{a^2b+ab^2}{\dfrac{(a+b)}{2}\times a\times (\dfrac{b}{a})^{1/3}\times a\times (\dfrac{b}{a})^{2/3}}\\\\\implies \dfrac{2ab(a+b)}{(a+b)(a^2\times (\dfrac{b}{a})^{1/3+2/3})}\\\\\implies \dfrac{2ab}{a^2\times \dfrac{b}{a}}\\\\\implies \dfrac{2ab}{ab}\\\\\implies 2$

Answer is 2 .

Last note :

n th term of a G.P is arⁿ⁻¹ . Hope you know indices rules . I am not writing them ...

Answer 2

Solution:

Let a and b are two positive numbers,

i)it is given that $\alpha$ is arithmetic mean of a and b.

$\alpha = \frac{a+b}{2}$

$\implies 2\alpha = a+b$-----(1)

and

ii) $\beta \:and \:\omega$

are geometric means of a and b.

Let r is the common ratio of the G.P

$\beta = ar$----(2)

$\omega = ar^{2}$---(3)

$b = ar^{3}$-----(4)

Substitute (5) in (2), we get

$a\left(\frac{\omega}{\beta}\right) = \beta$

$\implies a = \frac{\beta^{2})}{\omega}$----(6)

Substitute (5) and (6) in (4) , we get,

b=ar³

$b=\frac{\beta^{2}}{\omega}\times \frac{\omega^{3}}{\beta^{3}}$

$\implies b = \frac{\omega^{2}}{\beta}$----(7)

Now,

substitute (6) and (7) in (1), we get

$2\alpha = \frac{\beta^{2}}{\omega}+ \frac{\omega^{2}}{\beta}$

$2 = \frac{\beta^{3}+\omega^{3}}{\alpha\beta\omega}$

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