Page 371 - HOW TO PROVE IT: A Structured Approach, Second Edition

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Appendix 1: Solutions to Selected Exercises 357
(b) We use induction on n.
Base case: n = 1. This case is taken care of by part (a).
Induction step: Suppose n ≥ 1, and the arithmetic-geometric mean
n
inequality holds for lists of length 2 . Now let a 1 , a 2 ,..., a 2 n+1 be a
list of 2 n+1 positive real numbers. Let
n
n
a 1 + a 2 +· · · + a 2 n a 2 +1 + a 2 +2 + ··· + a 2 n+1
m 1 = and m 2 = .
2 n 2 n
n
n
Notice that a 1 + a 2 +· · · + a 2 = m 1 2 , and similarly a 2 +1 +
n
n
n
a 2 +2 +· · · + a 2 n+1 = m 2 2 . Also, by inductive hypothesis, we know
√ √
2 n
n
n
that m 1 ≥ a 1 a 2 ··· a 2 and m 2 ≥ 2 n a 2 +1 a 2 +2 ··· a 2 n+1. Therefore
n
n
a 1 + a 2 +· · · + a 2 n+1 m 1 2 + m 2 2 n m 1 + m 2 √
= = ≥ m 1 m 2
2 n+1 2 n+1 2

2 n √ 2 n √
≥ a 1 a 2 ··· a 2 n a 2 +1 a 2 +2 ··· a 2 n+1
n
n
2 n+1 √
= a 1 a 2 ··· a 2 n+1.
(c) We use induction on n.
Base case: If n = n 0 , then by assumption the arithmetic-geometric
mean inequality fails for some list of length n.
Induction step: Suppose n ≥ n 0 , and there are positive real numbers
a 1 , a 2 ,..., a n such that
a 1 + a 2 +· · · + a n √
< n a 1 a 2 ··· a n .
n
Let m = (a 1 + a 2 +· · · + a n )/n, and let a n+1 = m. Then we have
√ n
m < n a 1 a 2 ··· a n ,so m < a 1 a 2 ··· a n . Multiplying both sides of
this inequality by m gives us m n+1 < a 1 a 2 ··· a n m = a 1 a 2 ··· a n+1 ,
√
so m < n+1 a 1 a 2 ··· a n+1 . But notice that we also have mn = a 1 +
a 2 +· · · + a n ,so
a 1 +· · · + a n+1 mn + m m(n + 1) √
= = = m < n+1 a 1 a 2 ··· a n+1 .
n + 1 n + 1 n + 1
Thus, we have a list of length n + 1 for which the arithmetic-geometric
mean inequality fails.
(d) Suppose that the arithmetic-geometric mean inequality fails for some
list of positive real numbers. Let n 0 be the length of this list, and choose
n
an integer n ≥ 1 such that n 0 ≤ 2 . (In fact, we could just let n = n 0 ,
as you will show in exercise 12(a) in Section 6.3.) Then by part (b),
the arithmetic-geometric mean inequality holds for all lists of length