344. The formulas established in Arts. 340, 341, are fundamental; and since they contain the five elements, if any three of these are given, formulas may be deduced for finding the other two, as in the following The formulas for n, exhibited in the table for convenience of reference, require for their solution methods not yet given, but which will be presented in their proper place; and those containing the nth degree of the unknown quantity may present difficulty, unless n is less than 3. Hence, for the solution of questions in Geometrical Progression, we have the following RULE. Substitute in the appropriate formula the given quantities, and reduce the result. EXAMPLES. 1. If the first term of a geometrical progression is 7, the ratio 3, and the number of terms 5, what is the last term? Here, a = 7, r = = 3, and n = 5, to find l. 7 = 7 (3)5−1 = 7 × 3 = 567, Ans. 2. If the first term of a geometrical progression is 7, the ratio 3, and the number of terms 5, what is the sum of the series? Ans. 847. 3. If the ratio is 2, the number of terms 6, and the last term 128, what is the first term? Ans. 4. 4. If the first term is 2, the last term 4374, and the number of terms 8, what is the ratio? Ans. 3. and 128. Ans. 2, 8, 32. 5. Insert three geometrical means between 6. If the first term is 2, the ratio 4, and the number of terms 12, what are the last term, and the sum of the series? Ans. Last term, 8388608; sum of the series, 11184810. 7. Required the sum of the series 1, 3, ƒ, infinity. .... • Ans. 1. ૐ ૐ, to 11. Required the sum of 1}+1−1+ infinity. 12. A person who saved every year half as much again as he saved the previous year, had in seven years saved $2059. How much did he save the first year r? Ans. $64. 13. A gentleman, by agreement, boarded 9 days, paying 3 cents for the first day, 9 cents for the second day, 27 cents for the third day, and so on, in this ratio. Required the cost. Ans. $295.23. HARMONICAL PROGRESSION. 345. Three or more quantities are said to be in HARMONICAL PROGRESSION, when their reciprocals form an arithmetical progression. Thus, 1,,,, are in harmonical progression, because their reciprocals, 1, 3, 5, 7, . are in arithmetical progression. 346. If three consecutive terms of an Harmonical Progression be taken, the first has the same ratio to the third that the difference of the first and second has to the difference of the second and third. For, if a, b, c are in harmonical progression, their recip 347. From the preceding it follows that every series of quantities in harmonical progression may be easily converted into an arithmetical progression, and then the rules of the latter may be applied. There will be found, however, no general expression for the sum of an harmonical series. Thus, 1. Given the first two terms of an harmonical progression, to find the nth term. Let a and b be the first two terms, and the nth term ; 1 then and are the first two terms of an arithmetical a 1 ī progression, and is its nth term; therefore, Τ 2. To insert m harmonical means between a and 7. Here, if d be the common difference of the reciprocals of the terms, we have whence the arithmetical progression is found; and by inverting its terms, the harmonical means will be ascertained. EXAMPLES. 1. The first term of an harmonical progression is 1, and the second term ; find the fifth term. 2. Insert two harmonical means between and also between 4 and 5. 3 Ans.. and i Ans., 47, 413. 3. Continue the series 3, g, for two terms. Ans.. PROBLEMS REQUIRING THE APPLICATION OF THE PRINCIPLES OF THE PROGRESSIONS. 1. It is required to find four numbers in arithmetical progression, such that the product of the extremes shall be 45, and the product of the means 77. SOLUTION. Let x be the first term, and y the common difference; then the numbers will be x, x + y, x+2y, x+3y; and by the conditions, |