L, n, and S; any three of them being given, the other two can be determined. Two independent equations are sufficient to determine two unknown quantities, (Art. 45,) and it is immaterial which two are unknown if the other three are given. We perceive that the value of any letter, L for example, can be drawn from equation (B) as well as from (A). It can also be drawn from either of the equations after n or a is eliminated from them. Hence, the value of L may take four different forms. The same may be said of the other letters, and there being five quantities or letters and four different forms to each, the subject of arithmetical progression may include twenty different equations. But we prefer to make no display with these equations, believing they would add darkness rather than light, as they are all essentially included in the two equations, (A) and (B), and these can be remembered literally and philosophically, and the entire subject more surely understood. These two equations are sufficient for problems relating to arithmetical series, and we may use them without modification by putting in the given values just as they stand, and afterwards reducing them as numeral equations. EXAMPLES. 1. The sum of an arithmetical series is 1455, the first term 5, and the number of terms 30. What is the common difference? Ans. 3. Here S-1455, a=5, n=30. L and d are sought. Equation (B) 1455=(5+Z)15. Reduced L=92 Equation (A) 92=5+29d. Reduced d=3, Ans. 2. The sum of an arithmetical series is 567, the first term 7, and the common difference 2. What is the number of terms? Ans. 21. Here s 567, a=7, d=2. L and n are sought. 3. Find seven arithmetical means between 1 and 49. Observe that the series must consist of 9 terms. Hence, a=1, L=49, n=9. Ans. 7, 13, 19, 25, 31, 37, 43. 4. The first term of an arithmetical terms 280, the number of terms 32. What is the common difference, and the last term? Ans. d=1, L=161. series is 1, the sum of the 5. Insert three arithmetical means between and 1· Ans. The means are 3 5 1 I 6. Find nine arithmetical means between 9 and 109. 8' 12' 24 Ans. d--10. 7. What debt can be discharged in a year by paying 1 cent he first day, 3 cents the second, 5 cents the third, and so on, inreasing the payment each day by 2 cents? Ans. 1332 dollars 25 cents. 8. A footman travels the first day 20 miles, 23 the second, 26 he 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. 9. What is the sum of n terms of the progression of 1, 2, 3, 4, 5, &c.? Ans. S=2(1+n). 10. 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. 11. A man bought a certain number of acres of land, paying for the first, $; for the second, $3; and so on. When he came to settle he had to pay $3775. How many acres did he purchase, and what did it average per acre? Ans. 150 acres at $251 per acre. Problems in Arithmetical Progression to which the preceding formulas, (A) and (B), do not immediately apply. (Art. 117.) When three quantities are in arithmetical progression, it is evident that the middle one must be the exact mean of the three, otherwise it would not be arithmetical progression; therefore the sum of the extremes must be double of the mean. Take, for example, any three consecutive terms of a series, as a+2d, a+3d, a+4d; and we perceive by inspection that the sum of the extremes is double the mean. When there are four terms, the sum of the extremes is equal to the sum of the means, by (Art. 116.) To facilitate the solution of problems, when three terms are in question, let them be represented by (x-y), x, (x+y), y being the common difference. When four numbers are in question, let them be represented by (x-3y), (xy), (x+y), (x+3y); 2y being the common dif ference. So in general for any other number, assume such terms that the common difference will disappear by addition. 1. There are five numbers in arithmetical progression, the sum of these numbers is 65, and the sum of their squares is 1005. What are the numbers? Let x= the middle term, and y the common difference. Then x-2y, x-y, x, x+y, x+2y, will represent the numbers, and their sum will be 5x=65, or x=13. Also, the sum of their squares will be 5x+10=1005 or x+2y=201. But 22169; therefore, 2y2-32, y2=16 or y=4. Hence, the numbers are 13-8-5, 9, 13, 17 and 21. 2. There are three numbers in arithmetical progression, their sum is 18, and the sum of their squares 158. What are those numbers? Ans. 1, 6 and 11. 3. It is required to find four numbers in arithmetical progression, the common difference of which shall be 4, and their continued product 176985. Ans. 15, 19, 23 and 27. 4. There are four numbers in arithmetical progression, the sum of the extremes is 8, and the product of the means 15. What are the numbers? Ans. 1, 3, 5, 7. 5. A person travels from a certain place, goes 1 mile the first day, 2 the second, 3 the third, and so on; and in six days after, another sets out from the same place to overtake him, and travels uniformly 15 miles a day. How many days must elapse after the second starts before they come together? Reconcile these two answers. Ans. 3 days and 14 days. 6. A man borrowed $60; what sum shall he pay daily to cancel the debt, principal and interest, in 60 days; interest at 10 per cent. for 12 months, of 30 days each? Ans. $1 and of a cent. 7. There are four numbers in arithmetical progression, the sum of the squares of the extremes is 50, the sum of the squares of means is 34; what are the numbers? Ans. 1, 3, 5, 7. 8. The sum of four numbers in arithmetical progression is 24, their continued product is 945. What are the numbers? Ans. 3, 5, 7, 9. 9. A certain number consists of three digits, which are in arithmetical progression, and the number divided by the sum of its digits is equal to 26; but if 198 be added to the number its digits will be inverted. What is the number? Ans. 234. CHAPTER II. GEOMETRICAL PROGRESSION. (Art. 118.) When numbers or quantities differ from each other by a constant multiplier in regular succession, they constitute a geometrical series, and if the multiplier be greater than unity, the series is ascending; if it be less than unity, the series is descending. Thus, 2 6 18: 54: 162: 486, is an ascending series, the multiplier, called the ratio, being three; and 81: 27 : 9 : 3 : 1 : , &c., is a descending series, the multiplier or ratio being. Hence, a ar: ar2: ar3: ar1 : ar3: ar6 : &c., may represent any geometrical series, and if r be greater than 1, the series is ascending, if less than 1, it is descending. (Art. 119.) Observe that the first power of r 2d term, the 2d power in the 3d term, the third 4th term, and thus universally the power of the term is one less than the number of the term. The first term is a factor in every term. term of this general series is ar. The nth term would be ar11. stands in the power in the ratio in any Hence the 10th The 17th term would be ar16. Therefore, if n represent the number of terms in any series, and L the last term, then L-a-1 (1) (Art. 120.) If we represent the sum of any geometrical series by s, we have s=a+ar+ar2+ar3+ &c... ar22+ar"-1. Multiply this equation by r, and we have rs=ar+ar2+ar3+ &c. ar2¬1+ar". Subtract the upper from the lower, and observe that As these two equations are fundamental, and cover the whole subject of geometrical progression, let them be brought together for critical inspection. |