904 CHAPTER 12 Sequences; Induction; the Binomial Theorem In Problems 17–28, expand each expression using the Binomial Theorem. 17. ( ) +x 1 5 18. ( ) −x 1 5 19. ( ) −x 2 6 20. ( ) +x 3 5 21. ( ) +x3 1 4 22. ( ) +x2 3 5 23. ( ) + x y 2 2 5 24. ( ) − x y 2 2 6 25. ( ) + x 2 6 26. ( ) − x 3 4 27. ( ) + ax by 5 28. ( ) − ax by 4 In Problems 29–42, use the Binomial Theorem to find the indicated coefficient or term. 29. The coefficient of x6 in the expansion of ( ) +x 3 10 30. The coefficient of x3 in the expansion of ( ) −x 3 10 31. The coefficient of x7 in the expansion of ( ) −x2 1 12 32. The coefficient of x3 in the expansion of ( ) +x2 1 12 33. The coefficient of x7 in the expansion of ( ) +x2 3 9 34. The coefficient of x2 in the expansion of ( ) −x2 3 9 35. The 5th term in the expansion of ( ) +x 3 7 36. The 3rd term in the expansion of ( ) −x 3 7 37. The 3rd term in the expansion of ( ) −x3 2 9 38. The 6th term in the expansion of ( ) +x3 2 8 39. The coefficient of x0 in the expansion of ( ) + x x 1 2 12 40. The coefficient of x0 in the expansion of ( ) −x x 1 2 9 41. The coefficient of x4 in the expansion of ( ) −x x 2 10 42. The coefficient of x2 in the expansion of ( ) + x x 3 8 Applications and Extensions 43. Use the Binomial Theorem to find the numerical value of ( ) 1.001 5 correct to five decimal places. [Hint: ( ) ( ) = + − 1.001 1 10 5 3 5] 44. Use the Binomial Theorem to find the numerical value of ( ) 0.998 6 correct to five decimal places. 45. Show that − ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ = n n n 1 and ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ = n n 1. 46. Stirling’s Formula An approximation for n!, when n is large, is given by π( ) ( ) ≈ + − n n n e n ! 2 1 1 12 1 n Calculate 12!, 20!, and 25! on your calculator. Then use Stirling’s formula to approximate 12!, 20!, and 25!. 47. Challenge Problem If n is a positive integer, show that ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ + ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ + + ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ = n n n n 0 1 2n [Hint: ( ) = + 2 1 1 ; n n now use the Binomial Theorem.] 48. Challenge Problem If n is a positive integer, show that ( ) ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ − ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ + ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ − + − ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ = n n n n n 0 1 2 1 0 n 49. Challenge Problem Find the value of ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ + ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ + ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ + ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ + ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ + ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ 5 0 1 4 5 1 1 4 3 4 5 2 1 4 3 4 5 3 1 4 3 4 5 4 1 4 3 4 5 5 3 4 5 4 3 2 2 3 4 5 50. Challenge Problem Pascal Figures The entries in the Pascal triangle can, for ≥ n 2, be used to determine the number of k-sided figures that can be formed using a set of n points on a circle. In general, the first entry in a row indicates the number of n-sided figures that can be formed, the second entry indicates the number of ( ) −n 1 -sided figures, and so on. For example, if a circle contains 4 points, the row for = n 4 in the Pascal triangle shows the number of possible quadrilaterals (1), the number of triangles (4), and the number of line segments (6) that can be formed using the four points. (a) How many hexagons can be formed using 8 points lying on the circumference of a circle? (b) How many triangles can be formed using 10 points lying on the circumference of a circle? (c) How many dodecagons can be formed using 20 points lying on the circumference of a circle? 51. Challenge Problem Find the coefficient of x4 in ( ) ( ) ( ) ( ) = − + − + + − f x x x x 1 1 1 2 2 2 2 10 52. Challenge Problem In the expansion of [ ] ( ) + + a b c , 2 8 find the coefficient of the term containing a b c . 5 4 2 Retain Your Knowledge Problems 53–62 are based on previously learned material. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for subsequent sections, a final exam, or later courses such as calculus. 53. Solve = + 6 5 . x x 1 Express the answer both in exact form and as a decimal rounded to three decimal places. 54. For = + v i j 2 3 and = − w i j 3 2 : (a) Find the dot product ⋅ v w. (b) Find the angle between v and w. (c) Are the vectors parallel, orthogonal, or neither? 55. Solve the system of equations: − − = + + =− + − = ⎧ ⎨ ⎪⎪ ⎪⎪ ⎩ ⎪⎪ ⎪⎪ x y z x y z x y z 0 2 3 1 4 2 12
RkJQdWJsaXNoZXIy NjM5ODQ=