Question

Find the geometric mean between 6 and 48 given the sequence (3,6,12,24,48,96)

Answer 1

Answer:

16.97

Step-by-step explanation:

the geometric mean value of a set of n numbers is the nth root of the product of all n numbers.

so, here this means

$gm = \sqrt[6]{3 \times 6 \times 12 \times 24 \times 48 \times 96}$

this would be 16.97

but careful, the problem only asks for the gm between 6 and 48 of the sequence.

so, we actually only consider the subset 6, 12, 24, 48.

therefore

$gm = \sqrt[4]{6 \times 12 \times 24 \times 48}$

this is also

$gm = \sqrt[4]{3 \times 2 \times 3 \times 2 \times 2 \times 3 \times 2 \times 2 \times 2 \times 3 \times 2 \times 2 \times 2 \times 2}$

$= \sqrt[4]{ {3}^{4} \times {2}^{10} }$

$= \sqrt[4]{ {3}^{4} \times {2}^{4} \times {2}^{4} \times {2}^{2} }$

$= 3 \times 2 \times 2 \times \sqrt[4]{ {2}^{2} }$

$= 12 \times \sqrt{2}$

so, we could specify the result as that simple expression

or calculate it

gm = 16.97

hey, the result is the same as for the complete sequence.

coincidence ? no, it is not. but that is a subject for a different question.