| 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 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 |
| 10 ...... | 3 | + | 6 | + | 12 | + &c. ... | 847422675 |
| 7 | 49 | 201768035 | |||||
| 20 ...... | 1 | + | 1 | + | 1 | + &c. ... | 1048575 |
| 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 = 2b + c, where a is greater than 2b, the square of a, or aa, which is (68) 4bb + 4bc + cc is greater than 4bb, and so on.