Geometric mean of first group of 5 observations is 8 and that of second group of 4 observations is 1282. Then, grouped geometric mean is.

# Geometric mean of first group of 5 observations is 8 and that of second group of 4 observations is $128\sqrt{2}$. Then, grouped geometric mean is.

1. A

64

2. B

$32\sqrt{2}$

3. C

32

4. D

None of these

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### Solution:

Suppose that, G1 is the GM of first 5 observations = 8 and G2 is the GM of last 4 observations = $128\sqrt{2}$ Also, let G be the grouped mean

$\begin{array}{l}=\sqrt[9]{\left({\mathrm{G}}_{1}^{{\mathrm{n}}_{1}}×{\mathrm{G}}_{2}^{{\mathrm{n}}_{2}}\right)}=\sqrt[9]{\left({\mathrm{G}}_{1}^{{\mathrm{n}}_{1}}×{\mathrm{G}}_{2}^{{\mathrm{n}}_{2}}\right)}={\left({\mathrm{G}}_{1}^{5}×{\mathrm{G}}_{2}^{4}\right)}^{1/9}\\ ={\left[{8}^{5}×\left(128\sqrt{2}{\right)}^{4}\right]}^{1/9}={\left[{\left({2}^{3}\right)}^{5}×{\left({2}^{7+\frac{1}{2}}\right)}^{4}\right]}^{1/9}\\ ={\left({2}^{15}×{2}^{30}\right)}^{1/9}={2}^{\frac{45}{9}}={2}^{5}=32\end{array}$

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