Write the product of n geometric means between two number a and b

1.	Write the product of n geometric means between two number a and b
Answer» Let us suppose a and b are two numbers. Let us say G is the Geometric mean of a and b. ∴ a, G and b must be in Geometric Progression or GP. This means, common ratio = G/a = b/G Or, G₂ = ab Or, Gn = n(ab) ............ (1) Now, let us say G₁ , G₂ , G₃ ,.......G_n are n geomteric means between a and b. Which means that a , G₁ , G₂ , G₃ ...... G_n , b form a G.P. Note that the above GP has n+2 terms and the first term is a and last term is b, which is also the (n+2)^th term Hence, b = ar^n+2-1 where a is the first term. So, b = arⁿ⁺¹ ⇒ r = \(\bigg(\frac{b}{a}\bigg)^\frac{1}{n+1}\) Now the product of GP becomes Product = G₁G₂G₃......G_n = (ar)(ar²)(ar³)..(arⁿ) = aⁿ.r ^(1+2+3…+n) = aⁿ. r = aⁿ. r^{\(\frac{n(n+1)}{2}\)} Putting the value of r from equation 2 , we get = aⁿ. \(\bigg(\big(\frac{b}{a}\big)^\frac{1}{n+1}\bigg)^\frac{n(n+1)}{2}\) = (a.b)^{\(\frac{n}{2}\)}

Answer»

Let us suppose a and b are two numbers.

Let us say G is the Geometric mean of a and b.

∴ a, G and b must be in Geometric Progression or GP.

This means, common ratio = G/a = b/G

Or, G₂ = ab

Or, Gn = n(ab) ............ (1)

Now, let us say G₁ , G₂ , G₃ ,.......G_n are n geomteric means between a and b.

Which means that

a , G₁ , G₂ , G₃ ...... G_n , b form a G.P.

Note that the above GP has n+2 terms and the first term is a and last term is b,

which is also the (n+2)^th term

Hence, b = ar^n+2-1

where a is the first term.

So,

b = arⁿ⁺¹

⇒ r = \(\bigg(\frac{b}{a}\bigg)^\frac{1}{n+1}\)

Now the product of GP becomes Product = G₁G₂G₃......G_n

= (ar)(ar²)(ar³)..(arⁿ)

= aⁿ.r ^(1+2+3…+n)

= aⁿ. r

= aⁿ. r^{\(\frac{n(n+1)}{2}\)}

Putting the value of r from equation 2 , we get

= aⁿ. \(\bigg(\big(\frac{b}{a}\big)^\frac{1}{n+1}\bigg)^\frac{n(n+1)}{2}\)

= (a.b)^{\(\frac{n}{2}\)}

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