## Math 417 Homework 5: Due Friday February 21

*Instructions*: You can discuss these problems with others,
but write up your solutions on your own (i.e., don't just
copy someone else's solutions, else the feedback I give
you won't help you much). Please be neat and write in full sentences.

Do three of the last four problems, and one of the first three.

- [1] Problem #24 on p. 24. (Hint: see Example 10 on p. 16.)

- [2] Let {f
_{1}, f_{2}, f_{3}, ...}
be a sequence of integers such that f_{1} > 1 and
for n > 1 such that f_{n} > 3f_{n - 1} + 2.
Use induction to prove that f_{n} > 3^{n}/2 - 1 for
every integer n > 0. (Indicate which form of induction you used;
Theorem 0.4 or 0.5.)

- [3] Let {f
_{0}, f_{1}, f_{2}, f_{3}, ...}
be a sequence of integers such that 1 < f_{0} < f_{1} and
for n > 1 such that f_{n + 1} > f_{n} + f_{n - 1}.
Use induction to prove that f_{n} > 1.6^{n} for
every integer n > -1. (Indicate which form of induction you used;
Theorem 0.4 or 0.5.)

- [4] Problem 40 on p. 84.

- [5] Problem 54 on p. 85.

- [6] Problem 4 on p. 111.

- [7] Problem 8 on p. 111.