3. It is required to divide the number 25 into two such parts, that the sum of their square roots shall be 7. Ans. 16 and 9. 4. The product of a certain number, consisting of two places, by the sum of its digits, is 160, and if it be divided by 4 times the digit in unit's place, the quotient is 4; required the number. Ans. 32. 5. The difference between two numbers, multiplied by the greater, 16, but by the less, =12; required the numbers. Ans. 8 and 6. 6. Divide 10 into two such parts, that their product shall exceed their difference by 22. Ans. 6 and 4. 7. The sum of two numbers is 10, and the sum of their cubes is 370; required the numbers. Ans. 3 and 7. 8. The difference of two numbers is 2, and the difference of their cubes is 98; required the numbers. Ans. 5 and 3. 9. The sum of 6 times the greater of two numbers, and 5 times the less, is 50, and their product is 20; required the numbers. Ans. 5 and 4. 10. If a certain number, consisting of two places, is divided by the product of its digits, the quotient will be 2, and if 27 is added to it, the digits will be inverted; required the number. Ans. 36. 11. Find three such quantities, that the quotients arising from dividing the products of every two of them, by the one remaining, are a, b, and c. Ans. ±√ab, ±√ ac, and ±√bc. 12. The sum of two numbers is 9, and the sum of their cubes is 21 times as great as their sum; required the numbers. Ans. 4 and 5. 13. There are two numbers, the sum of whose squares exceeds twice their product, by 4, and the difference of their squares exceeds half their product, by 4; required the numbers. Ans. 6 and 8. 14. The fore wheel of a carriage makes 6 revolutions more than the hind wheel, in going 120 yards; but if the circumference of each wheel is increased 1 yard, it will make only 4 revolutions more than the hind wheel, in the same distance; required the circumference of each wheel. Ans. 4 and 5 yds. 15. Two persons, A and B, depart from the same place, and travel in the same direction; A starts 2 hours before B, and after traveling 30 miles, B overtakes A; but had each of them traveled half a mile more per hour, B would have traveled 42 miles before overtaking A. At what rate did they travel? Ans. A 2, and B 3 miles per hour. 16. A and B started at the same time, from two different points, toward each other; when they met on the road, it appeared that A had traveled 30 miles more than B. It also appeared, that it would take A 4 days to travel the road that B had come, and B 9 days to travel the road that A had come. Find the distance of A from B, when they set out. Ans. 150 miles. CHAPTER VIII. PROGRESSIONS AND PROPORTION. ARITHMETICAL PROGRESSION. ART. 220.—A series, is a collection of quantities or numbers, connected together by the signs + or —, and in which any one term may be derived from those which precede it, by a rule, which is called the law of the series. Thus, 1+3+5+7+9+, &c., are series; in the former of which, any term may be derived from that which precedes it, by adding 2; and in the latter, any term may be found by multiplying the preceding term by 3. ART. 221.-An Arithmetical Progression is a series of quantities which increase or decrease, by a common difference. Thus, the numbers 1, 3, 5, 7, 9, &c., form an increasing arithmetical progression, in which the common difference is 2. The numbers 30, 27, 24, 21, &c., form a decreasing arithmetical progression, in which the common difference is 3. REMARK. An arithmetical progression is termed, by some writers, an equidifferent series, or a progression by differences. Again, a, a+d, a+2d, a+3d, a+4d, &c., is an increasing arithmetical progression, whose first term is a, and common difference d. And if d be negative, it becomes a, a-d, a—2d, a—3d, a—4d, &c., which is a decreasing arithmetical progression, whose first term is a, and common difference d. ART. 222.-If we take an arithmetical series, of which the first term is a, and common difference d, we have 2d term 1st term +da+d 3d term =2d term +da+2d 4th term 3d term +d=a+3d, and so on. Hence, the coefficient of d in any term, is less by unity, than the number of that term in the series; therefore, the nth term =a+(n−1)d. If we designate the nth term by l, we have l=a+(n−1)d. RULE, FOR FINDING ANY TERM OF AN INCREASING ARITHMETICAL SERIES. Multiply the common difference by the number of terms less one, and add the product to the first term; the sum will be the required term. If the series is decreasing, then d is minus, and the formula is 1-a-(n-1)d. This gives the RULE, FOR FINDING ANY TERM OF A DECREASING ARITHMETICAL SERIES. Multiply the common difference by the number of terms less one, and subtract the product from the first term; the remainder will be the required term. EXAMPLES. 1. The first term of an increasing arithmetical series is 3, and common difference 5; required the 8th term. Here l, or 8th term =3+(8—1)5=3+35=38. Ans. 2. The first term of a decreasing arithmetical series is 50, and common difference 3; required the 10th term. Here l, or 10th term =50—(10-1)3-50-27-23. Ans. In the following examples, a denotes the first term, and d the common difference of an arithmetical series; d being plus when the series is increasing, and minus when it is decreasing. Ans. 28. Ans. 76. Ans. 103. Ans. 351⁄2 Ans. 5. Ans. 16. 3. a=3, and d=5; required the 6th term. 4. a=20, and d=4; required the 15th term. 5. a=7, and d=4; required the 16th term. 6. a=22, and 1=3; required the 100th term. 7. a=0, and d=2; required the 11th term.. 8. a=30, and d--2; required the 8th term. 9. a =—4, and d=3; required the 5th term. 10. a=- -10, and d=-2; required the 6th term. Ans. —20. 11. If a body falls during 20 seconds, descending 162 feet the first second, 48 feet the next, and so on, how far will it fall the twentieth second? Ans. 627 feet. Ans. 8. Of a REVIEW.-220. What is a series? Give examples. 221. What is an arithmetical progression? Give an example of an increasing series. decreasing series? 222. What is the rule for finding the last term of an increasing arithmetical series? Of a decreasing arithmetical series? Explain the reason of these rules. ART. 223.-Given, the first term a, the common difference d, and the number of terms n, to find s, the sum of the series. If we take an arithmetical series of which the first term is 3, common difference 2, and number of terms 5, it may be written in the following forms: 3. 3+2, 3+4, 3+6, 3+8 11, 11-2, 11-4, 11-6, 11-8. It is obvious, that the sum of all the terms in either of these lines, will represent the sum of the series; that is, s= 3+( 3+2)+( 3+4)+( 3+6)+( 3+8) And Adding, 2s 14+ 14 =14X5=70 Whence, s of 70-35. Now, let the last term, then writing the series both in a the series. 28=(1+a)+(1+a)+(l+a)+(l+a).. +(l+a) FOR FINDING THE SUM OF AN ARITHMETICAL SERIES. Multiply half the sum of the two extremes, by the number of terms. From the preceding, it appears, that the sum of the extremes is equal to the sum of any other two terms equally distant from the extremes. REMARK.-Since l=a+(n−1)d, if we substitute this in the place of l in the formula 8=( s=(1+a), it becomes 8= - ( 2a+(n−1)d)". This gives the following Rule, for finding the sum of an arithmetical series: To the double of the first term add the product of the number of terms less one, by the common difference, and multiply the sum by half the number of terms. REVIEW.-223. What is the rule for finding the sum of an arithmetical series? Explain the reason of the rule. EXAMPLES. 1. Find the sum of an arithmetical series, of which the first term is 3, last term 17, and number of terms 8. s= (3+17 8=80. Ans. 2 2. Find the sum of an arithmetical series, whose first term is 1, last term 12, and number of terms 12. Ans. 78. 3. Find the sum of an arithmetical series, whose first term is 0, common difference 1, and number of terms 20. Ans. 190. 4. Find the sum of an arithmetical series, whose first term is 3, common difference 2, and number of terms 21. 5. Find the sum of an arithmetical series, whose 10, common difference-3, and number of terms 10. Ans. 483. first term is A. -35. In this case, the sum of the negative terms exceeds that of the positive. ART. 224.-The equations la+(n−1)d and n s=(a+1), furnish the means of solving this general problem: Knowing any three of the five quantities a, d, n, l, s, which enter into an arithmetical series, to determine the other two. This question furnishes ten problems, the solution of which presents no difficulty; for we have always two equations, to determine the two unknown quantities, and the equations to be solved, are either those of the first or second degree. 1. Let it be required to find a in terms of l, n, and d. From the first formula, by transposing, we have a=l—(n−1)d. That is, the first term of an increasing arithmetical series is equal to the last term diminished by the product of the common difference into the number of terms less one. From the same formula, by transposing a, and dividing by n―1, 1-а we find d= n That is, in any arithmetical series, the common difference is equal to the difference of the extremes, divided by the number of terms less one. Examples, illustrating these principles, will be found in the collection at the close of this subject. REVIEW.-224. What are the fundamental equations of arithmetical progression, and to what general problem do they give rise? To what is the first term of an increasing arithmetical series equal? To what is the common difference of an arithmetical series equal? |