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N)4 35 7. 3n % 7n & 2 is an integral multiple of 8. * 8. 2@7n % 3@5n & 5 is an integral multiple of 24. * 9. x 2n & y 2n has x + y as a factor. 10. x 2n%1 % y 2n%1 has x % y as a factor. 11. For all integers n, prove the following: (a) 2n3 + 3n2 + n is an integral multiple of 6. (b) n5 - 5n3 + 4n is an integral multiple of 120. 12. Prove that n(n2 - 1)(3n + 2) is an integral multiple of 24 for all integers n. 13. Guess a formula for each of the following and prove it by mathematical induction: (a) 1 1 1 1 % % % ...

For n $ 1 can be expressed as k k. n k ' 1 In solving problems stated in terms of the sigma or pi notation, it is sometimes helpful to rewrite the expression in the original notation. 46 Problems for Chapter 6 1. Find each of the following: (a) The coefficient of x4y16 in (x + y)20. (b) The coefficient of x5 in (1 + x)15. (c) The coefficient of x3y11 in (2x - y)14. 2. Find each of the following: (a) The coefficient of a13b4 in (a + b)17. (b) The coefficient of a11 in (a - 1)16. (c) The coefficient of a6b6 in (a - 3b)12.

29. A 60-mile trip was made at 30 miles per hour and the return at 20 miles per hour. (a) How many hours did it take to travel the 120 mile round trip? (b) What was the average speed for the round trip? 30. Find x, given that 1/30, 1/x, and 1/20 are in arithmetic progression. What is the relation between x and the answer to Part (b) of problem 29? 31. Verify the factorization 1 - x7 = (1 - x)(1 + x + x2 + x3 + x4 + x5 + x6) and use it with x = 1/2 to find a compact expression for 1% 1 1 % 2 2 2 % 1 2 3 % 1 2 4 % 1 2 5 % 1 2 6 .