1873_solutions

Define n aj bn j1 for each n then the sequence b n

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Unformatted text preview: s for every positive integer n it follows from the principle of mathematical induction that P n is true for every positive integer n. 3. Given that x 1  2 and that the equation 88 x n 1  2  xn holds for every natural number n, prove that x n  2 for every natural number n. For each positive integer n we define P n to be the assertion that x n  2. The assertion P 1 says that 2  2 which is obviously true. Now suppose that n is any positive integer for which the statement P n is true. We see that x n 1  2  xn  2  2  2. Since P 1 is true and since the condition P n  P n1 holds for every positive integer n it follows from the principle of mathematical induction that P n is true for every positive integer n. 4. Given that x 1  2 and that the equation x n 1  2  xn holds for every natural number n, prove that x n  x n1 for every natural number n. For each positive integer n we define P n to be the assertion that x n  x n1 . The assertion P 1 says that 2 2 2 which is obviously true. Now suppose that n is any positive integer for which the statement P n is true. We see that x n 1  2  xn  2  x n 1  x n 2 . Since P 1 is true and since the condition P n  P n1 holds for every positive integer n it follows from the principle of mathematical induction that P n is true for every positive integer n. 5. Given that x 1  0 and that the equation 8x 31  6x n  1 n holds for every natural number n, prove the following two assertions: a. For every natural number n we have x n  1. For each positive integer n we define P n to be the assertion that x n  1. The assertion P 1 says that 0  1 which is true. Now suppose that n is any positive integer for which the statement P n is true. We see that 6x n  1  3 6  1  1. 8 8 Since P 1 is true and since the condition P n  P n1 holds for every positive integer n it follows from the principle of mathematical induction that P n is true for every positive integer n. x n 1  3 b. For every natural number n we have x n  x n1 . For each positive integer n we define P n to be the assertion that x n  x n1 . The assertion P 1 says that 0 1 2 which is obviously true. Now suppose that n is any positive integer for which the statement P n is true. We see that 6x n  1  3 6x n1  1  x . n 2 8 8 Since P 1 is true and since the condition P n  P n1 holds for every positive integer n it follows from the principle of mathematical induction that P n is true for every positive integer n. x n 1  3 6. Prove that every nonempty finite set of real numbers has a largest member. For each positive integer n we define P n to be the assertion that any set of real numbers that has exactly n members must have a largest member. Since the only member of any singleton is clearly the largest member of that singleton we know that the assertion P 1 is true. Now suppose that n is any positive integer for which the assertion P n is true. To prove that the assertion P n1 must also be true we suppose that S is a s...
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