Prove that the product of n geometric means between two quantities is

1.	Prove that the product of n geometric means between two quantities is equal to the nth power of a geometric mean of those two quantities.
Answer» Let us suppose a and b are two numbers. Let us say G is a number that is the Geometric mean of a and b Therefore a, G and b must be in Geometric Progression or GP. This means, common ratio = G/a = b/G Or, G² = ab Or, G = 1/2(ab)... (1) Now, let us say G₁, G₂, G₃, .......G_n are n geometric 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 the 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 = (b/a)^1/n+1 ....(2) Now the product of GP becomes Product = G₁G₂G₃......G_n = (ar)(ar²)(ar³).....(arⁿ) = aⁿ r^{(1+2+3+4+.............+n)} = aⁿ r^n(1+n)/2 Putting the value of r from equation 2 , we get = aⁿ (b/a)ⁿ(1+n)^/2(n+1) = (ab)^n/2 = (1/2ab)ⁿ Now, putting the value from equation 1, we get, Product = Gⁿ Or, G₁G₂G₃......G_n = Gⁿ

Prove that the product of n geometric means between two quantities is equal to the nth power of a geometric mean of those two quantities.

Answer»

Let us suppose a and b are two numbers.

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

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

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

Or, G² = ab

Or, G = 1/2(ab)... (1)

Now, let us say G₁, G₂, G₃, .......G_n are n geometric 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 the 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 = (b/a)^1/n+1 ....(2)

Now the product of GP becomes

Product = G₁G₂G₃......G_n

= (ar)(ar²)(ar³).....(arⁿ)

= aⁿ r^{(1+2+3+4+.............+n)}

= aⁿ r^n(1+n)/2

Putting the value of r from equation 2 , we get

= aⁿ (b/a)ⁿ(1+n)^/2(n+1)

= (ab)^n/2

= (1/2ab)ⁿ

Now, putting the value from equation 1, we get,

Product = Gⁿ

Or, G₁G₂G₃......G_n = Gⁿ