Solution:
Suppose that the five terms are a1, a2, a3, a4, a5.
where A = 32/9, B = 81/2
The above five terms are between A and B. So the GP is: A, a1, a2, a3, a4, a5, B.
Therefore, there are 7 terms in GP with the first term equal to 32/9 and the seventh term equalt to 81/2.
We know that, Tn = arn–1
Here, Tn = 81/2, a = 32/9 and
$ 81/2\text{ }=\text{ }32/9{{r}^{7-1}} $
$ \left( 81\times 9 \right)/\left( 2\times 32 \right)\text{ }=\text{ }{{r}^{6}} $
$ r\text{ }=\text{ }3/2 $
$ {{a}_{1}}~=\text{ }Ar\text{ }=\text{ }\left( 32/9 \right)\times 3/2\text{ }=\text{ }16/3 $
$ {{a}_{2}}~=\text{ }A{{r}^{2}}~=\text{ }\left( 32/9 \right)\times 9/4\text{ }=\text{ }8 $
$ {{a}_{3}}~=\text{ }A{{r}^{3}}~=\text{ }\left( 32/9 \right)\times 27/8\text{ }=\text{ }12 $
$ {{a}_{4}}~=\text{ }A{{r}^{4}}~=\text{ }\left( 32/9 \right)\times 81/16\text{ }=\text{ }18 $
$ {{a}_{5}}~=\text{ }A{{r}^{5}}~=\text{ }\left( 32/9 \right)\times 243/32\text{ }=\text{ }27 $
∴ The five GM between 32/9 and 81/2 are 16/3, 8, 12, 18, 27