Page 312 - HOW TO PROVE IT: A Structured Approach, Second Edition

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298 Mathematical Induction
Find a formula for a n and prove that your formula is correct. (Hint:
Imitate exercise 8.)
11. A sequence a 0 , a 1 , a 2 , ... is defined recursively as follows:
a 0 = 0;
a 1 = 1;
a 2 = 1;
1 3 1
for every n ≥ 3, a n = a n−3 + a n−2 + a n−1 .
2 2 2
th
Prove that for all n ∈ N, a n = F n , the n Fibonacci number.
12. For each positive integer n, let A n ={1, 2,..., n}, and let P n =
{X ∈ P (A n )| X does not contain two consecutive integers}. For exam-
ple, P 3 ={∅, {1}, {2}, {3}, {1, 3}}; P 3 does not contain the sets {1, 2},
{2, 3}, and {1, 2, 3} because each contains at least one pair of consecutive
integers. Prove that for every n, the number of elements in P n is F n+2 , the
th
(n + 2) Fibonacci number. (For example, the number of elements in P 3 is
5 = F 5 . Hint: Which elements of P n contain n? Which don’t? The answers
to both questions are related to the elements of P m , for certain m < n.)
13. Suppose n and m are integers and m > 0.
(a) Prove that there are integers q and r such that n = mq + r and
0 ≤ r < m. (Hint: If n ≥ 0, then this follows from Theorem 6.4.1.
If n < 0, then start by applying Theorem 6.4.1 to −n and m.)
(b) Prove that the integers q and r in part (a) are unique. In other

words, show that if q and r are integers such that n = mq + r and

0 ≤ r < m, then q = q and r = r .

(c) Prove that, as claimed in Section 3.4, every integer is either even or
odd but not both.
14. Suppose k is a positive integer. Prove that there is some positive integer
∗
k
n
a such that for all n > a,2 ≥ n . (In the language of exercise 16 of
k
n
Section 5.1, this means that if f (n) = n and g(n) = 2 then f ∈ O(g).
Hint: By the division algorithm, for any natural number n there are nat-
ural numbers q and r such that n = kq + r and 0 ≤ r < k. Therefore
n
q k
2 ≥ 2 kq = (2 ) . To choose a, figure out how large q has to be to guar-
q
antee that 2 ≥ n. You may find Example 6.1.3 useful.)
15. (a) Suppose k is a positive integer, a 1 , a 2 ,. . . , a k are real numbers, and
+
f 1 , f 2 , ... , f k , and g are all functions from Z to R. Also, suppose
that f 1 , f 2 , ... , f k are all elements of O(g). (See exercise 16 of
Section 5.1 for the meaning of the notation used here.) Define f :
Z → R by the formula f (n) = a 1 f 1 (n) + a 2 f 2 (n) +· · · + a k f k (n).
+
Prove that f ∈ O(g). (Hint: Use induction on k, and exercise 16(c) of
Section 5.1.)