Therefore, by addition, 2S = (a+1)+(a+1)+(a+1)+...+(a+1). (3) Now equation (3) expresses the sum of n terms, each equal to (a+1); hence, 2S = n(a+1); and dividing by 2, we obtain the formula, n S = 1/2 (a+1). (B) 352. To insert any number of arithmetical means between two given terms. Let n' denote the number of means to be inserted. Then the number of terms in the completed series will be n'+2; and we shall have This value of n substituted in formula (4), (349), gives whence, n = : n'+2. Having the common difference, the means are readily obtained. contain in all five quantities, a, l, n, d, S, four of which enter each equation. Now if any three of these quantities be given, the other two may be found; for, if the values of the three given quantities be substituted in the formulas, there will result two equations containing only two unknown quantities. 1. The first term of an arithmetical series is 5, the common difference 3, and the number of terms 24. Find the last term, and the sum of the series. We have given, hence, by formula (4), and by formula (B), = a5, d= 3, n = 24; 2. Given a 15, d : = terms. -2, and S 60, to find the number of Substituting the given values in (4) and (B), we have Both values of n are possible; for there are two series answering to the given conditions, one having 6 terms, and the other 10; these aro and 15, 13, 11, 9, 7, 5, 3, 1, −1, −3, 15, 13, 11, 9, 7, 5. The sum of either series is 60. EXAMPLES FOR PRACTICE. 1. The first term of an arithmetical series is 7, the common difference 3, and the number of terms 36; find the last term. Ans. 112. 2. The first term of an arithmetical series is 275, the last term 5, and the number of terms 46; required the sum of the terms. Ans. 6440. 3. The sum of an arithmetical series is 156, the number of terms 8, and the common difference 5. Required the two extremes. 4. Find the sum of the terms in an arithmetical progression, knowing that the first term is 1, the common difference, and the number of terms 101. Ans. 2626. 2437 5. Required to find four arithmetical means between 7 and 37. Ans. 13, 19, 25, 31. 6. The first term of an arithmetical series is 3, the number of terms 60, and the sum of the terms 3720; required the common difference, and the last term. Ans. d= 2, l = 121. 7. What will be the sum of the series if 9 arithmetical means be inserted between 9 and 109 ? Ans. 649. 8. If three arithmetical means be inserted between 1 and 1, what will be the common difference? $= 361 2+ Ans. 24. 3015 9. What debt can be discharged in a year by paying I cent the first day, 3 cents the second, 5 cents the third, and so on, increasing the payment cach day by 2 cents? Ans. 1332 dollars 25 cents. 10. A footman travels the first day 20 miles, 23 the second, 26 the third, and so on, increasing the distance each day 3 miles. How many days must he travel at this rate to go 438 miles? Ans. 12. 11. Find the sum of n terms of the progression of 1, 2, 3, 4, 5, 6, Ans. S= = (1+~). 2 ..... 12. Find the sum of n terms of the progression 1, 3, 5, 7,..... Ans. Sn3. 13. The sum of the terms of an arithmetical series is 950, the common difference is 3, and the number of terms 25. What is the first term? Ans. 2. 14. A man bought a certain number of acres of land, paying for the first $; for the second, $; and so on. When he came to settle he had to pay $3775. How many acres did he purchase? Ans. 150 acres. 15. The 14th, 134th, and last terms of an arithmetical progression are 66, 666, and 6666, respectively. Required the number of terms. Ans. 1334. THE TEN CASES. 354. Given any three of the quantities, a, l, n, d, S, to find the other two. This problem will present ten cases, each giving rise to two formulas, making in all twenty different formulas, or four values for each letter. The results in each case may be obtained directly from the two fundamental equations, or those of any particular case may be derived from some preceding case, as is most convenient. The whole will be left as an exercise for the student. PROBLEMS IN ARITHMETICAL PROGRESSION 7 10 WHICH THE FORMULAS DO NOT IMMEDIATELY APPLY. 355. When in the conditions of a problem no three of the five parts, a, l, n, d, S, are directly given, the general formulas will not directly apply. It is usually necessary in such instances to represent the several terms of the series by means of two or more unknown quantities; and for this purpose there are two methods of notation. 1st. Let x denote the first term and y the common difference; thus, x, (x+y), (x+2y), (x+3y).... This method of notation, however, is seldom the most expedient. 2d. When the number of terms is odd, denote the middle term by r, and the common difference by y; then we shall have, for three terms, for five terms, (x−y), x, (x+y); (x-2y), (x-y), x, (x+y), (x+2y). And when the number of terms is even, represent the two middle terms by x-y and x+y respectively, 2y being the common difference; thus, (x—3y), (x—y), (x+y), (x+3y). The advantage of the second method is, that the sum of all the terms, or the sum and difference of two tern.s equidistant from the extremes, will each contain but a single unknown quantity. 1. There are three numbers in arithmetical progression; the sum of these numbers is 18, and the sum of their squares is 158. What are the numbers? Ans. 1, 6, 11. 2. There are five numbers in arithmetical progression; their sum is 65, and the sum of their squares 1005. What are the numbers? Ans. 5, 9, 13, 17, 21. 3. It is required to find four numbers in such that their common difference shall be product 176985. arithmetical progression, 4, and their continued Ans. 15, 19, 23, 27. |