To get the next power of two, you simply double the previous value. For powers of two, we multiply by 2. Geometric series, whilst not named as such by Euclid, were referred to as numbers in continued proportion. This may be obvious if we rewrite 125 as 5 cubed. The missing number is 25 or 5 squared.
Geometric series are important in music.
The lengths of string or tube that produce twelve equally spaced notes form a geometric series starting at the original length and ending with half that length. This suggests a method for working out the sum of the first n terms of a geometric series. From Euclid's book IX Proposition IX - 35. If as many numbers as we please are in continued proportion and there is subtracted from the second and the last numbers equal to the first, then as the excess of the second is to the first, so will the excess of the last be to all those before it. As the excess of the second is to the first, so will the excess of the last be to all those before it' is talking about the ratio of the two differences found before to the first term and the sum of n terms respectively.