Find three geometric mean between 4/27 and 27

Think of this as a GP, where you want three terms between 4/27 and 27

27 = 4/27 * r^4
r^4 = 27^2 / 2^2
r = √(27/2)
Now just write the first 5 terms of the GP.

10 and 12

To find three geometric means between 4/27 and 27, we need to calculate the common ratio between consecutive terms and apply it three times to find the intermediate terms.

Step 1: Calculate the common ratio (r):
To find the common ratio (r) between two numbers, we divide the second number by the first number.
r = (27) / (4/27)

Step 2: Simplify the common ratio:
To simplify the common ratio, we multiply the numerator and denominator by 27.
r = (27 * 27) / 4 = 729 / 4

Step 3: Calculate the first geometric mean (GM1):
To find the first geometric mean (GM1), we take the square root of the product of the two given numbers.
GM1 = √[(4/27) * 27] = √(4) = 2

Step 4: Calculate the second geometric mean (GM2):
To find the second geometric mean (GM2), we multiply GM1 by the common ratio (r).
GM2 = GM1 * r = 2 * (729 / 4) = 3645 / 4

Step 5: Calculate the third geometric mean (GM3):
To find the third geometric mean (GM3), we multiply GM2 by the common ratio (r).
GM3 = GM2 * r = (3645 / 4) * (729 / 4)

Simplifying GM3 is a bit complicated and involves multiplication of large numbers, so the final result is:

GM3 = 244140625 / 65536

To find three geometric means between two numbers, follow these steps:

Step 1: Identify the first term (a) and the second term (b).

In this case, the first term is 4/27 and the second term is 27.

Step 2: Calculate the ratio (r).
To find the ratio, divide the second term (b) by the first term (a):
r = b/a

In this case, the ratio (r) is 27/(4/27) which can be simplified as r = (27*27)/4 = 182.25

Step 3: Calculate the first geometric mean (g1).

To find the first geometric mean, multiply the first term (a) by the square root of the ratio (r):
g1 = a * √r

In this case, g1 = (4/27) * √182.25

Step 4: Calculate the second geometric mean (g2).

To find the second geometric mean, multiply the first geometric mean (g1) by the square root of the ratio (r):
g2 = g1 * √r

In this case, g2 = g1 * √182.25

Step 5: Calculate the third geometric mean (g3).

To find the third geometric mean, multiply the second geometric mean (g2) by the square root of the ratio (r):
g3 = g2 * √r

In this case, g3 = g2 * √182.25

Now we can substitute the values and calculate g1, g2 and g3.

g1 = (4/27) * √182.25
g2 = g1 * √182.25
g3 = g2 * √182.25

By performing the calculations, you will get the three geometric means between 4/27 and 27.