_rrrr_ - 1 1 - _rrrr_ ---------- × _a_, or ---------- × _a_, _r_ - 1 1 - _r_ according as _r_ is greater or less than unity. The first fraction is _arrrr_ - _a_ _e_ - _a_ -------------, or (192) ---------. _r_ - 1 _r_ - 1 _a_ - _e_ Similarly, the second is ---------. 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 10 terms of the series 1 + 3 + 9 + 27 + &c. is required. The eleventh term is 59049, and ⁽⁵⁹⁰⁴⁹ ⁻ ¹⁾/₍₃₋₁₎ is 29524. Again, the sum of 18 terms of the series 2 + 1 + ½ + ½ + &c. of which the nineteenth term is ¹/₁₃₁₀₇₂, is 1 2 - ------ 131072 131070 ----------- = 3 ------. 1 - ½ 131072 EXAMPLES. 9 terms of 1 + 4 + 16 + &c. are 87381 6 12 847422675 10 ...... 3 + --- + ---- + &c. ... --------- 7 49 201768035 1 1 1 1048575 20 ...... --- + --- + --- + &c. ... ------- 2 4 8 1048576 195. The powers of a number or fraction greater than unity increase; for since 2½ is greater than 1, 2½ × 2½ is 2½ taken more than once, that is, is greater than 2½, and so on. This increase goes on without limit; that is, there is no quantity so great but that some power of 2½ is greater. To prove this, observe that every power of 2½ is made by multiplying the preceding power by 2½, or by 1 + 1½, that is, by adding to the former power that power itself and its half. There will, therefore, be more added to the 10th power to form the 11th, than was added to the 9th power to form the 10th. But it is evident that if any given quantity, however small, be continually added to 2½, the result will come in time to exceed any other quantity that was also given, however great; much more, then, will it do so if the quantity added to 2½ be increased at each step, which is the case when the successive powers of 2½ are formed. It is evident, also, that the powers of 1 never increase, being always 1; thus, 1 × 1 = 1, &c. Also, if _a_ be greater than _m_ times _b_, the square of _a_ is greater than _mm_ times the square of _b_. Thus, if _a_ = 2_b_ + _c_, where _a_ is greater than 2_b_, the square of _a_, or _aa_, which is (68) 4_bb_ + 4_bc_ + _cc_ is greater than 4_bb_, and so on. 196. The powers of a fraction less than unity continually decrease; thus, the square of ⅖, or ⅖ × ⅖, is less than ⅖, being only two-fifths of it. This decrease continues without limit; that is, there is no quantity so small but that some power of ⅖ is less. For if