Question

If g1, g2, g3 are three geometric means between two positive numbers a, b then find g1 .g3​

Answer 1

Given :-

• g₁ and g₂ and g₃ are three geometric means between two positive numbers a , b

To Find :-

• The value of g₁.g₃

Solution :-

The given mumbers are in GP. So it comes out to be

• m, g₁ , g₂ , g₃ , n.

Observing the sequence ,

• m = first term = a₁ (let)
• g₁ = second term = a₂
• g₂ = thrid term = a₃
• g₃ = fourth term = a₄
• b = Fifth term = a₅

In a Geometric progression ,

$\\ : \implies \sf \: \frac{a_2}{a_1} = \frac{a_3}{a_2} = \frac{a_4}{a_3} = \frac{a_5}{a_4} = ...= r \: (common \: ratio)$

Substituting the values of a₁ , a₂ , a₃ , a₄ and a₅ we get ;

$\\ : \implies \sf \: \frac{g_1}{a_1} = \frac{g_2}{g_1} = \frac{g_3}{g_2} = \frac{b}{g_3}$

Considering these two equations ,

$\\ :\implies \sf \frac{g_1}{a_1} = \underbrace{\frac{g_2}{g_1} = \frac{g_3}{g_2} }= \frac{b}{g_3}$

By cross multiplication ,

$\\ : \implies \sf \frac{g_1}{a_1} = \underbrace{\frac{g_2}{g_1} = \frac{g_3}{g_2} }= \frac{b}{g_3}$

$\\ : \implies \sf \: \frac{g_2}{g_1} = \frac{g_3}{g_2}$

$\\ : \implies{\underline{\boxed{\pink {\mathfrak{g_1.g_3 = {g_2}^{2} }}}}} \: \bigstar$

Hence ,

• The value of g₁.g₃ is g₂²