1080 CHAPTER 11 Further Topics in Algebra Find the indicated term of each binomial expansion. See Example 5. 49. sixth term of 14h - j28 50. eighth term of 12c - 3d214 51. seventeenth term of 1a2 + b222 52. twelfth term of 12x + y2216 53. fifteenth term of 1x - y3220 54. tenth term of 1a3 + 3b211 Concept Check Work each problem. 55. Find the middle term of 13x7 + 2y328. 56. Find the two middle terms of 1-2m-1 + 3n-2211. 57. Find the value of n for which the coefficients of the fifth and eighth terms in the expansion of 1x + y2n are the same. 58. Find the term(s) in the expansion of A3 + 2xB 11 that contain(s) x4. Relating Concepts For individual or collaborative investigation (Exercises 59–62) The factorial of a positive integer n can be computed as a product. n! = 1 # 2 # 3 # g# n Calculators and computers can evaluate factorials very quickly. Before the days of modern technology, mathematicians developed Stirling’s formula for approximating large factorials. The formula involves the irrational numbers p and e. n! ?!2Pn # nn # e−n As an example, the exact value of 5! is 120, and Stirling’s formula gives the approximation as 118.019168 with a graphing calculator. This is “off” by less than 2, an error of only 1.65%. Work Exercises 59–62 in order. 59. Use a calculator to find the exact value of 10! and its approximation, using Stirling’s formula. 60. Subtract the smaller value from the larger value in Exercise 59. Divide it by 10! and convert to a percent. What is the percent error to three decimal places? 61. Repeat Exercises 59 and 60 for n = 12. 62. Repeat Exercises 59 and 60 for n = 13. What seems to happen as n gets larger? 11.5 Mathematical Induction ■ Principle of Mathematical Induction ■ Proofs of Statements ■ Generalized Principle of Mathematical Induction ■ Proof of the Binomial Theorem Principle of Mathematical Induction Many statements in mathematics are claimed true for every positive integer. Any of these statements could be checked for n = 1, n = 2, n = 3, and so on, but because the set of positive integers is infinite, it would be impossible to check every possible case. For example, let Sn represent the statement that the sum of the first n positive integers is n1n + 12 2 . Sn: 1 + 2 + 3 + g+ n = n1n + 12 2
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