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 |
b = ar c = br = arr d = cr = arrr
and so on; whence the series
| a | b | c | d | &c. | ||||
|---|---|---|---|---|---|---|---|---|
| is | a | ar | arr | arrr | &c. | |||
| Hence | a | : | c | ∷ | a | : | arr | |
| (186) | ∷ | aa | : | aarr | ||||
| ∷ | aa | : | bb |
because,