| 1 | : | 2 | ∷ | 2 | : | 4 | | 2 | : | 2 | ∷ | 2 | : | 2 |
| 3 | 3 | 9 |
| 2 | : | 4 | ∷ | 4 | : | 8 | 2 | : | 2 | ∷ | 2 | : | 2 |
| 3 | 9 | 9 | 27 |
| &c. | | &c. |
192. Let a, b, c, d, e be in continued proportion; we have then
| a : b ∷ b : c | or | a | = | b | or | ac = bb |
| b | c |
| b : c ∷ c : d | | b | = | c | | bd = cc |
| c | d |
| c : d ∷ d : e | | c | = | d | | ce = dd |
| d | e |
Each term is formed from the preceding, by multiplying it by the same number. Thus,
| b | = | b | × a (180); c = | c | × b; |
| a | b |
| and since | a | = | b | , | b | = | c |
| b | c | a | b |
| or c = | b | × b. |
| a |
| Again, d = | d | × c , |
| c |
| but | d | = | c | , which is = | b | ; |
| c | b | a |
| therefore, d = | b | × c, and so on |
| a |
| If, then, | b | (which is called the common ratio of the series) |
| a |
| be denoted by r, we have |