Write the product of n geometric means between two number a and b
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 ,.......Gn are n geomteric means between a and b.
Which means that
a , G1 , G2 , G3 ...... Gn , 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 = arn+2-1`
where a is the first term.
So,
b = arn+1
⇒ ...(2)
Now the product of GP becomes
Product = G1G2G3......Gn
= (ar)(ar2)(ar3)..(arn)
= an.r(1+2+3…+n)
=
Putting the value of r from equation 2 , we get
=
= .