4. Required two geometrical means between 24 and 192. Ans. 48, 96. 5. Required 7 geometrical means between 3 and 768. Ans. 6, 12, 24, 48, 96, 192,384. 6. Find the value of 1++++..., to infinity. 16 Ans. 4. 7. Find the value of +1+&+2% +.... to infinity. Ans. 41. 8. Find the value of 5++++.... to infinity. Ans. 7. 9. Find the value of the decimal .323232.... to infinity. Ans. 10. Find the value of the decimal .212121.... to infinity. 32 Ans. 33. 11. Find the value of -+- 18+ 32 - .... to infinity. Ans.. 12. Find the value of +113 - 623 + ....to infinity. 15. The sum of a geometrical series is 1785, the ratio 2, and the number of terms 8; find the first term. Ans. 7. 16. The sum of a geometrical series is 7812, the ratio 5, and the number of terms 6; find the last term. Ans. 6250. 17. The first term of a geometrical series is 5, the last term 1215, and the number of terms 6. What is the ratio? Ans. 3. 18. A man purchased a house with ten doors, giving $1 for the first door, $2 for the second, $4 for the third, and so on. What did the house cost him? Ans, $1023. PROBLEMS IN GEOMETRICAL PROGRESSION TO WHICH THE FORMULAS DO NOT IMMEDIATELY APPLY. 365. The terms of a geometrical progression are represented in a general manner as follows: x, xy, xy, xy,.... In the solution of problems, however, the following notation is generally preferable: 1st. When the number of terms is odd, the series may be sented thus: repre 1. The sum of three numbers in geometrical progression is 26, and the sum of their squares 364. Let the numbers be denoted by x, Then What are the numbers? V xy, y. and x2+xy+y2 =3 = 364 = b. (2) Transposing Vxy in (1), squaring and reducing, 2. The sum of four numbers in geometrical progression is 15 or a, and the sum of their squares 85 or b. What are the numbers? notation for an even number of terms, we have Taking the proper ¡ x2+y= s3—3sp. Substituting the values of (x+y) and (x2+y), in (1) and (2), Squaring (3), and then transposing 2xy, or 2p, whence, from (4) and (5), (a—s)2—2p = b—s3+2p ; or, a2-2as+2s2-4p = b. (6) Clearing (5) of fractions, and putting xy=p in second member, x+y=ap-ps; or, s3-3sp=ap—ps ; whence, Substituting this value of p in (6), and reducing, we have or, a'-2as ab+2bs; Restoring the numerical values of a and b, whence, 15s+85870X15, s = 6. Substituting the values of a and s in (9), and we obtain 3. There are three numbers in geometrical progression; their sum is 21, and the sum of their squares is 189. Find the numbers. Ans. 3, 6, 12. 4. Divide the number 210 into three parts, so that the last shall exceed the first by 90, and the parts be in geometrical progression. Ans. 30, 60, and 120. ху, хүч 5. The sum of four numbers in geometrical progression is 30; and the last term divided by the sum of the mean terms is 1. What are the numbers? x. xy. xy, xy3 Ans. 2, 4, 8, and 16. 6. The sum of the first and third of four numbers in geometrical progression is 148, and the sum of the second and fourth is 888. What are the numbers?xxy.xyxy Ans. 4, 24, 144, and 864. 7. It is required to find three numbers in geometrical progression, '/.3. such that their sum shall be 14, and the sum of their squares 84. Ans. 2, 4, and 8. 8. There are four numbers in geometrical progression, the sec ond of which is less than the fourth by 24; and the sum of the What are the num Ans. 1, 3, 9, and 27. 9. There are three numbers in geometrical progression; the sum of the first and second is 20, and the difference of the second and third is 30. What are the numbers? X.,kyAns. 5, 15, 45. 10. The continued product of three numbers in geometrical progression is 216, and the sum of the squares of the extremes is 328. What are the numbers? Ans. 2, 6, 18. 11. The sum of three numbers in geometrical progression is 13, and the sum of the extremes being multiplied by the mean, the product is 30. What are the numbers What are the numbers. Y Ans. 1, 3, and 9. 12. There are three numbers in geometrical progression; their continued product is 64, and the sum of their cubes is 581. What are the numbers? X, xy, xyz Ans. 2, 4, 8. 13. There are three numbers in geometrical progression; their continued product is 1, and the difference of the first and second is What are Ans. 1, 1, 5. to the difference of the second and third as 5 to one. x-3y, x-y1 x+yq* The su The sum of 120 dollars was divided between four persons in 44-120 such a manner that the shares were in arithmetical progression; if X=30 the second and third had each received 12 dollars less, and the fourth 24 dollars more, the shares would have been in geometrical X-24, x+7-121 by, x-4-12, x + 7-ndssion! Find the shares. progression Ans. $3, $21, $39, and $57. 17 118 - 4: : 18 - 7:15 15. +7 There are three numbers in geometrical progression, whose sum is 31, and the sum of the first and last is 26. What are the numbers? Ans. 1, 5, and 25. 16. The sum of six numbers in geometrical progression is 189, and the sum of the second and fifth is 54. What are the numbers? Ans. 3, 6, 12, 24, 48, and 96. (da) 17. The sum of six numbers in geometrical progression is 189, and the sum of the two means is 36. What are the numbers? Ans. 3, 6, 12, 24, 48, and 96. 18. A man borrowed p dollars; what sum must he pay yearly in order to cancel the debt in n years, interest being allowed on the unpaid parts of the principal at r cents per annum on a dollar? x = Ans. pr(1+r)n (1+r)”—1 dollars. IDENTICAL EQUATIONS. 366. An Identical Equation is one in which the two members are either the same algebraic expression, or the one member is merely another form for the other. In every case, either the one member may be reduced to the other directly, or the two members may be reduced to some expression different from either, from which both members may be supposed to originate. Thus, are identical equations. In the first, the two members have exactly |