In a series in GP if two terms are 6 and 30.How many mimimum terms would be required between 6 and 30 so that the terms added between 6 and 30 are...

Let us take a =a1 = 6 as the starting term, an = 30 as the
nth term and r the common ratio between the consecutive
terms.

ar^(n-1) = 30.

If there
are no terms , between 6 and 30, then the sum is unique= 6+30 =36
unique.

If there are 1 term a2 between 6 and 30, then r =
x/6 = 30/x.

Therefore x = sqrt(30*6) = +6sqrt5 Or
-6sqrt5.

So there are are two GPs: 6, -6sqrt5 , 30 with r =
-sqrt5 and

6 , 6sqrt5 , 30 with common ratio r =
+sqrt5.

Therefore the GP is not
unique.

When there are 0 or even number of terms between a1
and an ,or  between 6 and 30, then n  is even . Then an/a1 = r^(n-1) = 30/6 = 5 has a
single real solution  as n-1 is odd.

So the minimum number
of terms beteen 6 and 30 is zero for the sum to be
unique.

Also n should be even and not odd sothat the common
ratio is unique which leads to the unique sum.