Solution:
Let a1, a2, a3, a4, a5, a6 be the six terms
We have, A = 27, B = 1/81
According to the question, these 6 terms are between A and B. So the GP is given by:
A, a1, a2, a3, a4, a5, a6, B.
So there are 8 terms in GP with the first term equal to 27 and eighth terms equal to 1/81.
We know that the expression:
Tn = arn–1
Here, Tn = 1/81, a = 27 and
$ 1/81\text{ }=\text{ }27{{r}^{8-1}} $
$ 1/\left( 81\times 27 \right)\text{ }=\text{ }{{r}^{7}} $
$ r\text{ }=\text{ }1/3 $
$ {{a}_{1}}~=\text{ }Ar\text{ }=\text{ }27\times 1/3\text{ }=\text{ }9 $
$ {{a}_{2}}~=\text{ }A{{r}^{2}}~=\text{ }27\times 1/9\text{ }=\text{ }3 $
$ {{a}_{3}}~=\text{ }A{{r}^{3}}~=\text{ }27\times 1/27\text{ }=\text{ }1 $
$ {{a}_{4}}~=\text{ }A{{r}^{4}}~=\text{ }27\times 1/81\text{ }=\text{ }1/3 $
$ {{a}_{5}}~=\text{ }A{{r}^{5}}~=\text{ }27\times 1/243\text{ }=\text{ }1/9 $
$ {{a}_{6}}~=\text{ }A{{r}^{6}}~=\text{ }27\times 1/729\text{ }=\text{ }1/27 $
∴ The six GM between 27 and 1/81 are 9, 3, 1, 1/3, 1/9, 1/27