Geometric mean

For positive a and b, the number √(ab) is called their geometric mean (G.M.). For 2 and 8 it is 4, and 2, 4, 8 follow one another in a G.P. with ratio 2. For 4 and 9 it is 6.

Any number of terms can be inserted between two positive numbers a and b so that the whole list is a G.P. With n numbers G₁, …, Gₙ inserted, b is term n + 2, so b = arⁿ⁺¹ and r = (b/a)^(1/(n + 1)).

Then G₁ = ar, G₂ = ar², …, Gₙ = arⁿ.

Between 1 and 256, three numbers give r⁴ = 256, r = 4 (real positive root): 4, 16, 64. Between 2 and 162, three numbers give r⁴ = 81, r = 3: 6, 18, 54.

Taking the negative real root r = −3 gives the other list −6, 18, −54, which is also a G.P. from 2 to 162.