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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 Fall '08
 STAFF
 Math, Calculus

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