3. Substituting the Dividing the first couplet of (2) by x2, we obtain (3); from (3) we form (4), which reduced gives (6), or y value of y in (1), we obtain (7), or x = 4. = Ans. 4, 12, 36. 2. The sum of three numbers in geometrical progression is 39, and the sum of their squares 819. What are the numbers? SOLUTION. Let x, xy, and y represent the series. Then x + √xy + y = 39 (1) x2 + xy + y2=819 (2) x−√xy+y=21 (3) 2x+2y=60 (4) x+y=30 (5) 2xy=18 (6) xy=81 (7) Dividing (2) by (1), we obtain (3); adding (3) to (1), we obtain (4), which reduced gives (5); subtracting (3) from (1), we obtain (6), which reduced gives (7). Combining (5) and (7) as the sum and product are combined in Example 1, Art. 188, we obtain x = 27 and y = 3. Ans. 3, 9, 27. 3. Of four numbers in geometrical progression the dif ference between the fourth and second is 60; and the sum of the extremes is to the sum of the means as 13 : 4. What are the numbers? SOLUTION. Let x, xy, xy2, and xy represent the series. Then Dividing the first couplet of (2) by xy + x, we obtain (3); from (3) we form (4), which reduced gives (6), or y = 4. Substituting this value of y in (1) and reducing, we obtain (8), or x = = 1. Ans. 1, 4, 16, 64. 4. Of four numbers in geometrical progression the sum of the first two is 10 and of the last two 160. What are the numbers ? Ans. 2, 8, 32, 128. 5. A man paid a debt of $310 at three payments. The several amounts paid formed a geometrical series, and the last payment exceeded the first by $240. several payments? What were the Ans. $10, $50, $250. 6. In the series x, xy, and y what is the ratio? 8. There are four numbers in geometrical progression whose continued product is 64; and the sum of the series is to the sum of the means as 5: 2. What are the numbers? Ans. 1, 2, 4, 8. 9. There are five numbers in geometrical progression; the sum of the first four is 156, and the sum of the last four 780. What are the numbers? 10. There are three numbers in geometrical progression whose sum is 126; and the sum of the extremes is to the mean as 17: 4. What are the numbers? 11. The sum of the squares of three numbers in geometrical progression is 2275; and the sum of the extremes is 35 more than the mean. What are the numbers? 12. Of four numbers in geometrical progression the sum of the first and third is 52; and the difference of the means is to the difference of the extremes as 5: 31. What are the numbers? SECTION XXIV. MISCELLANEOUS EXAMPLES. 1. From 6ac5ab+c2 take 3ac-{3ab - (c- c2) Ans. 3ac2ab+2c2+6 c. +7c}. - y(xy-x2)}) to its simplest form. ху x2 Ans. 2 x2y+205. 3. Reduce (a − b + c)2 — (a (c — a — b) — { 6 (a + b + c) — c (a − b — c)}) to its simplest form. 5. Reduce (a2b2) c — (a — b) {a (b+c) — b (a—c)} to its simplest form. Ans. 0. — 6. Reduce (a+b) x − (b −c) c — { (b — x) b − (b −c) (b + c)} -ax to its simplest form. 7. Multiply a +2a2b-3ab2 by (3 a3ba2 b2). 8. Multiply a+6a2+9 by a 6a29. 11. Divide 118x281 x by 1+6x+9x2. 12 Divide 9 a2 + 1-4 at 6a by 12 a2 - 3 a. x6 13. Divide 9 x3 — 7 x2 y2 + 2 y3 by 3x+2x2y - y2. 14. Divide 23 a 307 a3 +6 at by 3a-2 a2 — 5. 15. Find the prime factors of a* — ba. 16. Find the prime factors of 4 m n2 17. Find the prime factors of x2 49 m2 n1o. 2xy + y2. ys. 18. Find the prime factors of x3 19. Find the greatest common divisor of 5 x3 +15ys and 4x3 + 8 x2 y + 8 x y2 + 4 y3. 10 x2 y Ans. xy. 20. Find the greatest common divisor of 8 a b5 + 24 a ba + 16 a b and 7 bo + 7 b5 + 7 ba — 7 b2. Ans. bb. 21. Find the greatest common divisor of 6 x2 + 7 x y 3y2 and 12x2+22 x y + 6 y2. 22. Find the greatest common divisor of 4 x + 4x3 40 to its lowest terms. 23. Reduce 29. Reduce to one fraction with the least possible de |