understood (or translated into words) by those who are acquainted with the common signs that are employed in Algebra*. *To explain the methods of obtaining the following Theorems, as well as those in geometrical progression, belongs rather to Algebra than to Arithmetic; they are, therefore, purposely omitted in this work, which only embraces subjects immediately connected with Arithmetic. EXERCISES. 1. Required the sum of an arithmetical series, consisting of 100 terms; the first term being 3, and the common difference 4? 2. The first term of a decreasing arithmetical series is 420, the common difference 3, and the number of terms 50; required the sum of the series? 3. The greatest term of a decreasing arithmetical series is 23, the common difference, and the number of terms 39; required the sum of the series? 4. The sum of an arithmetical series is 14820, the first term 0, and the last 297; required the number of terms, and common difference? 5. The last term of an arithmetical series is 505, the first 5, and common difference 5; required the sum of the series, and number of terms? 6. A person agrees to pay a debt in a year, by weekly payments; to pay Is the first week, 2s. the next, 3s. the next, &c. required the debt, and last payment? 7. How many strokes do the clocks of Venice, which go on to 24 o'clock, strike in a day? 8. A trader, while he continued in business, spent £4680. The first year he had spent £30, and he had increased his expenses yearly by £12. How long was he in trade? 9. A trader spent £60, the ninth year he was in business; his expenses had yearly increased, during that period, by an equal sum, and were in all £432. What did he spend the year, and what was the yearly increase? first GEOMETRICAL PROGRES SION. WHEN a rank, or series of numbers, increase by the same multiplier, or decrease by the same divisor, they form a geometrical progression*. Thus, 2, 4, 8, 16, 32, 64, &c. is an increasing geometrical progression, in which the common multiplier is 2. The first and last terms are usually called the extremes, and the constant multiplier the ratio. In every geometrical progression there are five different par ticulars concerned: viz. 1. The least term. 2. The greatest term. 3. The number of terms. 5. The sum of all the terms. Any three of these being given, the other two may be found. As questions in geometrical progression are most easily performed by Logarithms, and can only be performed by those who have acquired a knowledge of the first principles of Algebra, the theorems for solving most of the cases are here inserted, in * 1. If the progression consist of three terms, the product of the two extremes is equal to the square of the mean term. Example; 4, 8, 16, is a geometrical progression; therefore, 16×4=82. 2. If it consist of four terms, the product of the two extremes is equal to the product of the two means. Example: 4, 8, 16, 32, is a geometrical progression; therefore, 32×4=16×8. 3. In a geometrical progression consisting of any number of terms, the product of the two extremes is equal to the product of any two terms that are equally distant from them, or to the square of the mean term, when the number of terms are odd. Example: let the prog. be 2, 4, 8, 16, 32, 64, 128; then 128×2=162=256=32×8. |