8. Find the formula for the value of d, when a, n, and 9. Find the formula for the value of d, when a, l, and S are given. Ans. d= (l + a) (la) 28-(1+a) S 10. Find the formula for the value of d, when n, 1, and S are given. 2 (nl―S) Ans. d= n (n − 1) 11. Find the formula for the value of a, when d, n, and 12. Find the formula for the value of a, when d, l, and 226. To find any one of the five elements when three others are given. RULE. Substitute the given values in that formula whose first member is the required term, and whose second contains the three given terms. 1. Given d = 2, 7 = 21, and S120, to find a. In Example 12, Art. 225, we find (1), the required formula; substituting the given values of d, l, and S, we obtain (2), which reduced gives (3), or a = 3, or - - 1. 6. Find the 100th term of the series 3, 10, 17, &c. Ans. 696. 7. Find the sum of 100 terms of the series 3, 10, 17, & c Ans. 34950. 8. Find the common difference and sum of the series whose first term is 25, last term 95, and number of terms 15. Ans. d = 5; S=900. 9. Find the sum of the natural series of numbers from 1 to 100, inclusive. 10. Find the sum of 10 of the odd numbers 1, 3, 5, &c. 11. Find the sum of 10 of the even numbers 2, 4, 6, &c. 12. How many strokes does a clock strike in 12 hours? 13. If 100 trees stand in a straight line 10 feet from one another, how far must a person, starting from the first tree and returning to it each time, travel to go to every tree? Ans. 18 miles. 14. If a person should save a cent the first day, two cents the second, three the. third, and so on, how much would he save in 365 days? Ans. $667.95. 15. If a person should save $25 a year and put this sum at simple interest at 5 per cent at the end of each year, to how much would it amount at the end of 25 years? PROBLEMS TO WHICH THE FORMULAS DO NOT DIRECTLY APPLY. 227. Sometimes in examples in progression the terms are not directly given, but are implied in the conditions of the problem. In this case the formulas cannot be directly used, but the terms can be represented by unknown quantities, and equations formed according to the given conditions. 228. If x = first term and y = the common difference; then x, x + y, x+2y, x+3y, &c. will represent the series. It will often be found more convenient when the number of terms is odd to represent the middle term by x and the common difference by y; then the series for three terms will be and when the number of terms is even, to represent the two middle terms by x y and xy, and the common difference by 2y; then the series for four terms is x-3y, x-y, x + y, x + 3y. The advantage of this latter method is, that the sum of the series, or the sum or difference of any two terms equally distant from the mean, or means, will contain but one unknown quantity. 1. The sum of three numbers in arithmetical progression is 15, and the sum of their squares is 83. What are the numbers? Let x the mean term and y = the common difference; then the series will be x — y, x, and x+y. 2. The sum of four numbers in arithmetical progression is 44, and the sum of the cubes of the two means is 2926. Ans. 5, 9, 13, 17. 3. Find seven numbers in arithmetical progression such that the sum of the first and fifth shall be 10, and the difference of the squares of the second and fourth 40. 4. There are four numbers in arithmetical progression ; the product of the first and third is 20; and the product of the second and fourth 84. What are the numbers? Ans. 2, 6, 10, 14. 5. The sum of four numbers in arithmetical progression is 32; and their product 3465. What are the numbers? Ans. 5, 7, 9, 11. 6. The sum of the squares of the extremes of four numbers in arithmetical progression is 461; and the sum of the squares of the means 425. What are the numbers? 7. A certain number consists of three figures which are in arithmetical progression; if the number is divided by the sum of its figures, the quotient will be 15; and if 396 is added to the number, the order of the figures will be inverted. What is the number? Aus. 135. 8. Find four numbers in arithmetical progression such that the sum of the squares of the first and third shall be 104, and of the second and fourth 232. 9. Find four numbers in arithmetical progression such that the sum of the squares of the first and second shall be 29, and of the third and fourth 185. SECTION XXIII. GEOMETRICAL PROGRESSION. 229. A GEOMETRICAL PROGRESSION is a series in which each term, except the first, is derived by multiplying the preceding term by a constant quantity called the ratio. 230. If the first term is positive, when the ratio is a positive integral quantity, the series is called an ascending series, and when the ratio is a positive proper fraction a descending series; but if the first term is negative, the series is ascending when the ratio is a positive proper frac tion, and descending when the ratio is a positive integral quantity. Thus, If the ratio is negative, the terms of the progression are alternately positive and negative. Thus, if the ratio -2 and the first term 3, the series will be The positive terms of these two series constitute an ascending progression whose ratio is the square of the given ratio; and the negative terms a descending progression having the same ratio. 231. In Geometrical Progression there are five elements, any three of which being given, the other two can be found. These elements are the same as in Arithmetical Progression, except that in place of the common difference we have the ratio. |