Page 300 - HOW TO PROVE IT: A Structured Approach, Second Edition

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286 Mathematical Induction
5. Suppose r is a real number and r = 1. Prove that for all n ∈ N,
n n+1
r − 1
i
r = .
r − 1
i=0
(Note that this exercise generalizes Example 6.1.1 and exercise 7 of
Section 6.1.)
6. Prove that for all n ≥ 1,
∗
n
1 1
≤ 2 − .
i 2 n
i=1
7. (a) Suppose a 0 , a 1 , a 2 ,..., a n and b 0 , b 1 , b 2 ,..., b n are two sequences
of real numbers. Prove that
n n n

(a i + b i ) = a i + b i .
i=0 i=0 i=0
(b) Suppose c is a real number and a 0 , a 1 ,..., a n is a sequence of real
numbers. Prove that
n n

c · a i = (c · a i ).
i=0 i=0
8. The harmonic numbers are the numbers H n for n ≥ 1 defined by the
∗
formula
n
1
H n = .
i
i=1
(a) Prove that for all natural numbers n and m,if n ≥ m then H n − H m ≥
n−m . (Hint: Let m be an arbitrary natural number and then proceed
n
by induction on n, with n = m as the base case of the induction.)
(b) Prove that for all n ≥ 0, H 2 ≥ 1 + n/2.
n
(c) (For those who have studied calculus) Show that lim n→∞ H n =∞,
so ∞ 1 diverges.
i=1 i
9. Let H n be defined as in exercise 8. Prove that for all n ≥ 2,
n−1

H k = nH n − n.
k=1
n
10. Find a formula for (i · (i!)) and prove that your formula is correct.

i=1
11. Find a formula for n i and prove that your formula is correct.
i=0 (i+1)!
12. (a) Prove that for all n ∈ N, 2 > n.
∗ n
n 2
(b) Prove that for all n ≥ 9, n! ≥ (2 ) .
2
(c) Prove that for all n ∈ N, n! ≤ 2 (n ) .