b being ar, bb is arar or aarr. Again,
| a | : | d | ∷ | a | : | arrr | ||
|---|---|---|---|---|---|---|---|---|
| (186) | ∷ | aaa | : | aaarrr | ||||
| ∷ | aaa | : | bbb | |||||
| a | : | e | ∷ | aaaa | : | bbbb | , and so on; |
that is, the first bears to the nᵗʰ term from the first the same proportion as the nᵗʰ power of the first to the nᵗʰ power of the second.
- A short rule may be found for adding together any number of terms of a continued proportion. Let it be first required to add together the terms 1, r, rr, &c. where r is greater than unity. It is evident that we do not alter any expression by adding or subtracting any numbers, provided we afterwards subtract or add the same. For example,
p = p - q + q - r + r - s + s
Let us take four terms of the series, 1, r, rr, &c. or,
1 + r + rr + rrr
It is plain that
rrrr - 1 = rrrr - rrr + rrr - rr + rr - r + r - 1
Now (54), rr - r = r ( r -1), rrr