7. The sum of a certain number of terms of an A. P. is 45, and the first and last of these terms are 1 and 17 respectively. Find the number of terms and the common difference of the series. (ii.) √12x-5 + √3 x − 1 = √27x-2. 9. Find the value of the seventh term in the expansion of (a + x)" when a = 1, x = }, n = 9. 10. A man starting from A at 11 o'clock passed the fourth milestone at 11.30 and met another man (who started from B at 12) at 12.48; the second man passed the fourth milestone from A at 1.40: find the distance between A and B, and the second man's rate. 11. Show that x3 + 13 a2x> 5 ax2 + 9 a3, if x > a. 12. Extract the cube root of 44 x8 +63 x2 + x6 +27 + 6x5 + 21 x2 + 54 x. 15. Two vessels, one of which sails 2 miles an hour faster than the other, start together upon voyages of 1680 and 1152 miles respectively; the slower vessel reaches its destination one day before the other: how many miles per hour did the faster vessel sail? 17. Two numbers are in the ratio 2:7; the numbers obtained by adding 6 to each of the given numbers are in the duplicate ratio of 2:3. Find the numbers. 18. Solve (i.) 2 bx2 + 2 b = 4x + b2x. 20. Find the sides of a rectangle the area of which is unaltered if its length be increased by 2 feet while its breadth is diminished by 1 foot, and which loses of its area if its length be diminished by 2 feet and its breadth by 4 feet. 21. The first term of a G. P. exceeds the second term by 1, and the sum to infinity is 81: find the series. 22. Find the number of permutations which can be made from all the letters of the word Mississippi. 23. Solve (i.) √x + 2 + √4 x + 1 − √9 x + 7 = 0. 24. Find the condition that one root of ax2 + bx + c = 0 shall be n times the other. 25. Find the value of x8 - 3x2 - 8x + 15 when x = 3 + i. = 26. Given log 648 2.81157, log 864 = 2.93651, find the logarithm of 3 and of 5. 27. Two trains run, without stopping, over the same 36 miles of rail. One of them travels 15 miles an hour faster than the other and accomplishes the distance in 12 minutes less. Find the speeds of the two trains. 28. Extract the square root of 9x42x3y +163 x2у2 — 2 xy3 + 9 ya, 29. Find, by logarithms, the value of 31. The men in a regiment can be arranged in a column twice as deep as its breadth; if the number be diminished by 206, the men can be arranged in a hollow square three deep having the same number of men in each outer side of the square as there were in the depth of the column; how many men were there at first in the regiment? 33. Simplify 8+ (2 × 4-5) - √2 ÷ 4 ̄† - (32). 34. A man bought a field the length of which was to its breadth as 8 to 5. The number of dollars that he paid for 1 acre was equal to the number of rods in the length of the field; and 13 times the number of rods round the field equalled the number of dollars that it cost. Find the length and breadth of the field. 35. Solve (i.) x2 + xy + 3y2 = 14+2 √2, 2x2 + xy + 5 y2 = 24 + 2 √2. (ii.) 2x+3y= 10, 36. Find two numbers whose sum added to their product is 34, and the sum of whose squares diminished by their sum is 42. 37. Find the sixth term in the expansion of each of the following expressions: 38. A varies directly as B and inversely as C; A = when B and C = 39. Solve find B when _A = √√48 and C = √75. (i.) Vx+12+ √x+12= 6. = 40. Form an equation whose roots shall be the arithmetic and harmonic means between the roots of x2 — px + q = 0. CHAPTER XXXVII. BINOMIAL THEOREM. 406. It may be shown by actual multiplication that (a + b)(a + c)(a + d) (a + e) = a*+ (b+c+d+e)a3 + (bc + bd + be + cd + ce+de)a2 +(bcd+bce+bde + cde)a + bcde (1) We may, however, write this result by inspection; for the complete product consists of the sum of a number of partial products each of which is formed by multiplying together four letters, one being taken from each of the four factors. If we examine the way in which the various partial products are formed, we see that (1) The term at is formed by taking the letter a out of each of the factors. (2) The terms involving as are formed by taking the letter a out of any three factors, in every way possible, and one of the letters b, c, d, e, out of the remaining factor. (3) The terms involving a2 are formed by taking the letter a out of any two factors, in every way possible, and two of the letters b, c, d, e, out of the remaining factors. (4) The terms involving a are formed by taking the letter a out of any one factor, and three of the letters b, c, d, e, out of the remaining factors. (5) The term independent of a is the product of all the letters b, c, d, e. Ex. Find the value of (a− 2)(a + 3) (a − 5) (a + 9). The product = a2 + (−2+3 − 5 + 9) a3 + ( − 6 + 10 − 18 — 15+ 27 −45)a2 = a + 5 a3 — 47 a2 — 69 a + 270. 407. If in equation (1) of the preceding article we sup pose c=d=e=b, we obtain (a + b) = a* + 4 a3b +6 a2b2 + 4 ab3 +ba. We shall now employ the same method to prove a formula known as the Binomial Theorem, by which any binomial of the form a + b can be raised to any assigned positive integral power. 408. To find the expansion of (a+b)" when n is a positive integer. Consider the expression (a + b)(a+c)(a + d) ..... (a + k), the number of factors being n. ... The expansion of this expression is the continued product of the n factors, a + b, a + c, a +d, a+k, and every term in the expansion is of n dimensions, being a product formed by multiplying together n letters, one taken from each of these n factors. The highest power of a is a", and is formed by taking the letter a from each of the n factors. The terms involving an-1 are formed by taking the letter a from any n-1 of the factors, and one of the letters b, c, d, ... k from the remaining factor; thus the coefficient of an-1 in the final product is the sum of the letters b, c, d,... k; denote it by S1. ... The terms involving a"-2 are formed by taking the letter a from any n 2 of the factors, and two of the letters b, c, d,... k from the two remaining factors; thus the coefficient of a"-2 in the final product is the sum of the products of the letters b, c, d, k taken two at a time; denote it by S2. And, generally, the terms involving an are formed by taking the letter a from any n r of the factors, and r of the letters b, c, d,... k from the r remaining factors; thus the coefficient of a"-" in the final product is the sum of the products of the letters b, c, d, ·k taken r at a time; denote it by S ... |