Unformatted text preview: k − 1)!
=
=
= k (k + 1)(k + 2)
(k − 1)!
(k − 1)!
(k$$$
− 1)!
$
The stipulation k ≥ 1 is there to ensure that all of the factorials involved are defined.
2. We proceed by induction and let P (n) be the inequality n! > 3n . The base case here is n = 7
and we see that 7! = 5040 is larger than 37 = 2187, so P (7) is true. Next, we assume that P (k )
is true, that is, we assume k ! > 3k and attempt to show P (k + 1) follows. Using the properties
of the factorial, we have (k + 1)! = (k + 1)k ! and since k ! > 3k , we have (k + 1)! > (k + 1)3k .
Since k ≥ 7, k + 1 ≥ 8, so (k + 1)3k ≥ 8 · 3k > 3 · 3k = 3k+1 . Putting all of this together, we
have (k + 1)! = (k + 1)k ! > (k + 1)3k > 3k+1 which shows P (k + 1) is true. By the Principle
of Mathematical Induction, we have n! > 3n for all n ≥ 7.
Of all of the mathematical animals we have discussed in the text, factorials grow most quickly. In
problem 2 of Example 9.4.1, we proved that n! overtakes 3n at n = 7. ‘Overtakes’ may be too
polite a word,...
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- Fall '13
- Wong
- Algebra, Trigonometry, Cartesian Coordinate System, The Land, The Waves, René Descartes, Euclidean geometry
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