An Inequality: |
| (1) | (1 + 1-3)(1 + 2-3)(1 + 3-3)...(1 + n-3) < 3 |
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Copyright © 1996-2012 Alexander Bogomolny
Mathematical induction is a reasonable method to apply to proving (1) "for all n". As is the case with another example, there does not appear to be an obvious way to make induction work for (1).
| (1) | (1 + 1-3)(1 + 2-3)(1 + 3-3)...(1 + n-3) < 3 |
However, a strengthened inequality
| (2) | (1 + 1-3)(1 + 2-3)(1 + 3-3)...(1 + n-3) < 3 - 1/n |
is easily amenable to an inductive argument. Denote the left-hand side of (2) as A(n). The verification of (2) for
For n = k+1, we have
| A(k+1) | = A(k)(1 + (k+1)-3) |
| < (3 - 1/k)(1 + 1/(k+1)3) | |
| = 3 - 1/k + 3/(k+1)3 - 1/[k(k+1)3] | |
| = 3 - ((k+1)3 - 3k + 1)/[k(k+1)3]. |
Our goal of proving (2) for n = k+1 will be achieved provided
The latter simplifies to
and further to
| (3) | k2 - k + 2 > 0. |
Since P(x) = x2 - x + 2 has no real roots and
We thus obtain an example of two problems - one weaker, the other stronger - of which the weaker problem is more difficult to prove than the stronger one. This fits into the observation that more general problem are often easier than their specific instances.
References
- T. Andreescu, B. Enescu, Mathematical Olympiad Treasures, Birkhäuser, 2004
Inequalities to prove:
- An Inequality for Grade 8
- An Extension of the AM-GM Inequality
- A Mathematical Rabbit out of an Algebraic Hat
- An Inequality With an Infinite Series
- An Inequality: 1/2 * 3/4 * 5/6 * ... * 99/100 less than 1/10
- A Low Bound for 1/2 * 3/4 * 5/6 * ... * (2n-1)/2n
- An Inequality: Easier to prove a subtler inequality
- Inequality with Logarithms
- An inequality: 1 + 1/4 + 1/9 + ... less than 2
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Copyright © 1996-2012 Alexander Bogomolny
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