Page 310 - HOW TO PROVE IT: A Structured Approach, Second Edition

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296 Mathematical Induction
P(n) is a statement about a natural number n, and suppose that
∀n[(∀k < nP(k)) → P(n)]. Let Q(n) be the statement ∀k < nP(k).
(a) Prove ∀nQ(n) ↔∀nP(n) without using induction.
(b) Prove ∀nQ(n)by ordinary induction. Thus, by part (a), ∀nP(n)is
true.
√
2. This exercise gives an alternative way of writing the proof that 2is
irrational. Use strong induction to prove that ∀q ∈ N[q > 0 →¬∃p ∈
√
N(p/q = 2)].
√
∗ 3. (a) Prove that 6 is irrational.
√ √
(b) Prove that 2 + 3 is irrational.
4. The Martian monetary system uses colored beads instead of coins. A blue
bead is worth 3 Martian credits, and a red bead is worth 7 Martian credits.
Thus, three blue beads are worth 9 credits, and a blue and red bead together
are worth 10 credits, but no combination of blue and red beads is worth
11 credits. Prove that for all n ≥ 12, there is some combination of blue
and red beads that is worth n credits.
5. Suppose that x is a real number, x = 0, and x + 1/x is an integer. Prove
n
n
that for all n ≥ 1, x + 1/x is an integer.
∗ th
6. Let F n be the n Fibonacci number. All variables in this exercise range
over N
n
(a) Prove that for all n, i=0 F i = F n+2 − 1.
n 2
(b) Prove that for all n, i=0 (F i ) = F n F n+1 .
n
(c) Prove that for all n, F 2i+1 = F 2n+2 .
i=0
(d) Find a formula for n F 2i and prove that your formula is correct.
i=0
th
7. Let F n be the n Fibonacci number. All variables in this exercise range
over N.
(a) Prove that for all m ≥ 1 and all n, F m+n = F m−1 F n + F m F n+1 .
(b) Prove that for all m ≥ 1 and all n ≥ 1, F m+n = F m+1 F n+1 −
F m−1 F n−1 .
2 2 2 2
(c) Prove that for all n,(F n ) + (F n+1 ) = F 2n+1 and (F n+2 ) − (F n ) =
F 2n+2 .
(d) Prove that for all m and n,if m | n then F m | F n .
(e) See exercise 18 of Section 6.3 for the meaning of the notation used
in this exercise. Prove that for all n ≥ 1,

2n − 2 2n − 3 2n − 4 n − 1

F 2n−1 = + + + ··· +
0 1 2 n − 1
n−1
2n − i − 2
=
i
i=0