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Recursion 287
13. Suppose k is a positive integer.
2n
2
(a) Prove that for all n ∈ N,(k + n)! ≥ k .
n
2
(b) Prove that for all n ≥ 2k , n! ≥ k . (Hint: Use induction, and for the
base case use part (a). Note that in the language of exercise 16 of
n
Section 5.1, this shows that if f (n) = k and g(n) = n!, then f ∈
O(g).)
14. Prove that for every real number a and all natural numbers m and
m n
n, (a ) = a mn .
∗ 15. A sequence a 0 , a 1 , a 2 , ... is defined recursively as follows:
a 0 = 0;
for every n ∈ N, a n+1 = 2a n + n.
n
Prove that for all n ∈ N, a n = 2 − n − 1.
16. A sequence a 0 , a 1 , a 2 , ... is defined recursively as follows:
a 0 = 2;
2
for every n ∈ N, a n+1 = (a n ) .
Find a formula for a n and prove that your formula is correct.
17. A sequence a 1 , a 2 , a 3 , ... is defined recursively as follows:
a 1 = 1;
a n
for every n ≥ 1, a n+1 = .
a n + 1
Find a formula for a n and prove that your formula is correct.
n

18. For n ≥ k ≥ 0, the quantity is defined as follows:
∗
k
n n!

= .
k k! · (n − k)!
n
n

(a) Prove that for all n ∈ N, = = 1.
0 n
n
n+1 n
(b) Prove that for all n ≥ k > 0, = + .
k k k−1
(c) If A is a set and k ∈ N, let P k (A) be the set of all subsets of A
that have k elements. Prove that if A has n elements and n ≥ k ≥ 0,
n

then P k (A) has elements. (Hint: Prove by induction that ∀n ∈
k
n

N∀A[A is a set with n elements →∀k(n ≥ k ≥ 0 → P k (A) has
k
elements)]. Imitate exercises 10 and 11 of Section 6.2. In fact, this
exercise generalizes exercise 11 of Section 6.2. This exercise shows
n

that is the number of ways of choosing k elements out of a set of
k
size n, so it is sometimes called n choose k.)
(d) Prove that for all real numbers x and y and every natural number n,
n
n
n n−k k
(x + y) = x y .
k
k=0