308. If four quantities are in continued proportion, the first is to the fourth as the cube of the first is to the cube of the second. Note. The ratio a2: 62 is called the duplicate ratio, and the ratio a3: b3 the triplicate ratio, of a: b. 2. If y(x+z)2: (y+z)2, · (x + z)2: (y + z)2, prove that z is a mean proportional between x and y. From the given proportion, by Art. 293, Therefore z is a mean proportional between x and y. 3. Find the first term of the proportion whose last three terms are 18, 6, and 27. 4. Find the second term of the proportion whose first, third, and fourth terms are 4, 20, and 55. 5. Find a fourth proportional to, , and 4. 6. Find a third proportional to and §. 7. Find a mean proportional between 8 and 18. 8. Find a mean proportional between 14 and 42. 9. Find a mean proportional between 24 and 1⁄2· Solve the following equations: 10. 2x-5:3x+2=x-1: 7x+1. 11. 2-4-9= x2-5x+6: x2+4x+3. 13. (x:y= 3:5. 14. Sx+y: x-y=a+b: a-b. 15. Find two numbers in the ratio of 21 to 2, such that when each is diminished by 5, they shall be in the ratio of 1 to 1. 16. Divide 50 into two parts such that the greater increased by 3 shall be to the less diminished by 3, as 3 to 2. 17. Divide 12 into two parts such that their product shall be to the sum of their squares as 3 to 10. 18. Find two numbers in the ratio of 4 to 9, such that 12 is a mean proportional between them. 19. The sum of two numbers is to their difference as 10 to 3, and their product is 364. What are the numbers? -- 20. If a-b: b − c = b; c, prove that b is a mean proportional between a and c. 21. If 5a4b: 9a+2b=5b+4c: 96+2c, prove that b is a mean proportional between a and c. 22. If (a+b+c+d) (a−b-c+d) = (ab+cd) (a+b―c-d), prove that a:bc: d. 23. If ax-by: cx- dy = ay — bz: cy — dz, prove that y is a mean proportional between x and z. 24. Find two numbers such that if 3 be added to each, they will be in the ratio of 4 to 3; and if 8 be subtracted from each, they will be in the ratio of 9 to 4. 25. There are two numbers whose product is 96, and the difference of their cubes is to the cube of their difference as 19 to 1. What are the numbers? 26. Divide $564 between A, B, and C, so that A's share may be to B's in the ratio of 5 to 9, and B's share to C's in the ratio of 7 to 10. 27. A railway passenger observes that a train passes him, moving in the opposite direction, in 2 seconds; whereas, if it had been moving in the same direction with him, it would have passed him in 30 seconds. Compare the rates of the two trains. 28. Each of two vessels contains a mixture of wine and water. A mixture, consisting of equal measures from the two vessels, contains as much wine as water; and another mixture, consisting of four measures from the first vessel and one from the second, is composed of wine and water in the ratio of 2 to 3. Find the ratio of wine to water in each vessel. 29. Divide a into two parts such that the first increased by b shall be to the second diminished by b, as a +36 is to a-3b. XXVII. ARITHMETICAL PROGRESSION. 310. An Arithmetical Progression is a series of terms, each of which is derived from the preceding by adding a constant quantity called the common difference. ... Thus 1, 3, 5, 7, 9, 11, is an increasing arithmetical progression, in which the common difference is 2. 12, 9, 6, 3, 0, -3, ... is a decreasing arithmetical progression, in which the common difference is - 3. 311. Given the first term, a, the common difference, d, and the number of terms, n, to find the last term, l. The progression is a, a+d, a+2d, a+3d, ... It will be observed that the coefficient of d in any term is one less than the number of the term. Hence, in the nth, or last term, the coefficient of d will be n - 1. l = a + (n − 1)d That is, (I.) 312. Given the first term, a, the last term, l, and the number of terms, n, to find the sum of the series, S. ... S=a+(a + d) + (a + 2d) + ··· + (1 − d) +l. Writing the series in reverse order, S=1+(la) +(-2d) + ··· + (a + d) + a. ... Adding these equations, term by term, 28=(a+1)+(a+1)+(a+1)+ ··· + ( a +1)+(a+1) ... = = n(a + 1). S=12 (a+1). Therefore, (II.) 313. Substituting in (II.) the value of / from (I.), we have EXAMPLES. 314. 1. In the series 8, 5, 2, −1, −4, ... to 27 terms, find the last term and the sum. Substituting in (I.) and (II.), 7=8+(27 −1)(-3)=8—78=- 70. S − 27 (8 — 70) = 27 × (−31) — — 837. = 2 Note. The common difference may always be found by subtracting the first term from the second. Thus, in the series In each of the following, find the last term and the sum of |