193. 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_ -_rr_ = _rr_(_r_-1), _rrrr_-_rrr_ = _rrr_(_r_-1), and the above equation becomes _rrrr_ -1 = _rrr_(_r_-1) + _rr_ (_r_-1) + _r_ (_r_-1) + _r_-1; which is (54) _rrr_ + _rr_ + _r_ + 1 taken _r_-1 times. Hence, _rrrr_-1 divided by _r_-1 will give 1 + _r_ + _rr_ + _rrr_, the sum of the terms required. In this way may be proved the following series of equations: _rr_ - 1 1 + _r_ = -------- _r_ - 1 _rrr_ - 1 1 + _r_ + _rr_ = --------- _r_ - 1 _rrrr_ - 1 1 + _r_ + _rr_ + _rrr_ = ---------- _r_ - 1 _rrrrr_ - 1 1 + _r_ + _rr_ + _rrr_ + _rrrr_ = ----------- _r_ - 1 If _r_ be less than unity, in order to find 1 + _r_ + _rr_ + _rrr_, observe that 1 - _rrrr_ = 1 - _r_ + _r_ - _rr_ + _rr_ - _rrr_ + _rrr_ - _rrrr_ = 1 - _r_ + _r_(1 - _r_) + _rr_(1 - _r_) + _rrr_(1 - _r_); whence, by similar reasoning, 1 + _r_ + _rr_ + _rrr_ is found by dividing 1-_rrrr_ by 1-_r_; and equations similar to these just given may be found, which are, 1 - _rr_ 1 + _r_ = -------- 1 - _r_ 1 - _rrr_ 1 + _r_ + _rr_ = --------- 1 - _r_ 1 - _rrrr_ 1 + _r_ + _rr_ + _rrr_ = ---------- 1 - _r_ 1 - _rrrrr_ 1 + _r_ + _rr_ + _rrr_ + _rrrr_ = ----------- 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_. 194. 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