Chapter 5: Q31E (page 371)
Is the recursive or the iterative algorithm for finding the sequence in Exercise 29 more efficient?
The iterative algorithm is used for finding the sequence.
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Prove that whenever nis a nonnegative integer
Prove that if h > - 1 , then for all nonnegative integers n. This is called Bernoulli’s inequality.
Prove that the recursive algorithm for finding the sum of the first n positive integers you found in Exercise 8 is correct.
Use the well-ordering principle to show that if x and y are real numbers with x<y, then there is a rational number r with x<r<y. [Hint: Use the Archimedean property, given in Appendix 1, to find a positive integer A with . Then show that there is a rational number with denominator A between x and y by looking at the numbers , where is a positive integer.]
Suppose that you know that a golfer plays the first hole of
a golf course with an infinite number of holes and that if
this golfer plays one hole, then the golfer goes on to play
the next hole. Prove that this golfer plays every hole on
the course.
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