conty911 wrote:
If a and n are integers and n > 1, is n odd?
(1) a^(n–1) > a^n
(2) a^n > a^(3n)
If you factor, as in the post above, you can avoid cases if you arrange things differently. Rewriting Statement 1, noticing that a cannot be 1, we can factor out the smaller power, so can factor out a^(n-1) :
0 > a^n - a^(n-1)
0 > a^(n-1) (a - 1)
Now one term on the right side must be positive, and the other must be negative. But if a is a positive integer, that can't possibly be true -- both terms will be positive. So a must be negative, and (a-1) must be the negative term, and a^(n-1) must be the positive term. Since a is negative, that means the exponent must be even, so n-1 is even and n is odd.
Similarly for Statement 2, notice a cannot be -1, 0 or 1, and we have
0 > a^(3n) - a^n
0 > a^n [a^(2n) - 1]
and again, one term must be positive, the other negative. But a^(2n) has an even exponent, so it must be positive and so a^(2n) - 1 is the positive term (using the fact that a cannot be 0, 1 or -1). That makes a^n the negative term. But if a^n is negative, a must be negative, and n must be odd.
I don't think you need to do the problem algebraically, however. You can just imagine how different numbers would 'behave' in the inequalities given. So if you know that n is a positive integer greater than 1, and you know
a^(n-1) > a^n
then you might be able to see that a must be negative, since this inequality won't ever be true if a > 1. And if we know the base is negative, and we can see that one exponent is even and the other is odd (since the exponents are just consecutive integers), we know that one side of the inequality will become positive, and the other will remain negative. Clearly the larger side needs to be positive, which makes n-1 even, and n odd.
Similarly for the other Statement, the inequality
a^n > a^(3n)
simply cannot be true if a > 1. Since 1, 0 or -1 don't work in this inequality, a must be -2 or smaller. But then if n is even, it wouldn't matter if a is negative or positive, since even exponents just 'erase' our negative signs. And we know it must matter, since the inequality won't work if we have positive bases on either side. So the inequality will only work if n is odd so that the left and right sides both remain negative.
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