found, which are,
| 1 + r | = | 1 - rr |
|---|---|---|
| 1 - r | ||
| 1 + r + rr | = | 1 - rrr |
| 1 - r | ||
| 1 + r + rr + rrr | = | 1 - rrrr |
| 1 - r | ||
| 1 + r + rr + rrr + rrrr | = | 1 - rrrrr |
| 1 - r |
The rule is: To find the sum of n terms of the series, 1 + r + rr + &c., divide the difference between 1 and the (n + 1)ᵗʰ term by the difference between 1 and r.
- This may be applied to finding the sum of any number of terms of a continued proportion. Let a, b, c, &c. be the terms of which it is required to sum four, that is, to find a + b + c + d, or (192) a + ar + arr + arrr, or (54) a(1 + r + rr + rrr), which (193) is
| rrrr - 1 | × a , or | 1 - rrrr | × a , |
|---|---|---|---|
| r - 1 | 1 - r |
according as r is greater or less than unity. The first fraction is
| arrrr - a | , or (192) | e - a | . |
|---|---|---|---|
| r - 1 | r - 1 |
Similarly, the second is
| a - e | . |
|---|---|
| 1 - r |
The rule, therefore, is: To sum n terms of a continued proportion, divide the difference of the (n + 1)ᵗʰ and first terms by the difference between unity and the common measure. For example, the sum of