FIND THE TWO GEOMETRIC MEANS

Question

Find the two geometric means between √3 and 3.

Answer 1

This is easier to answer if you put your radicals into power with rational exponent

For √3 and 3, we will use 3^1/2 and 3^2/2

To get the geometric mean, multiply them by adding the exponents.

3^1/2 • 3^2/2 = 3^3/2

and then get the square root of the result (or raise it to 1/2 power).

(3^3/2)^1/2 = 3^3/4 or 27^1/4

Answer 2

To get two geometric means between 3^1/2 and 3^2/2 , think of these 4 terms as a geometric sequence where we want to find the value of "r".

We know that a_n = a₁r^n-1, i.e. to get from a₁ to the 4th term multiply by (4-1) factors of r.

So we have 3^2/2= 3^1/2r³.

r³= 3^2/2/ 3^1/2 = 3^1/2

So r = (3^1/2)^1/3= 3^1/6

To find those two geometric means:

a₁= 3^1/2

a₂=3^1/23^1/6=3^4/6

a₃=3^5/6

a₄=3^6/6

a₂ and a₃ are the two geometric means between √3¦and 3

Check here to see how to confirm:

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