If a and n are integers and n > 1, is n odd?

(1) a^(n–1) > a^n
(2) a^n > a^(3n)


Please provide me pointers to any similar questions, also if possible

Can some one explain steps to tackle such problems with least amount of time.For eg. how to select the values for plugging etc.
THanks
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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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From the conditions in both statements, we can deduce that a cannot be neither 0 nor 1.
A little algebraic manipulation can help in understanding the inequalities and also in choosing values for a.
Usually it is easier to compare a product of numbers to 0, so it is worth rearranging the inequalities such that there is a comparison to 0 involved.

(1) a^{n-1}>a^n can be rewritten as a^{n-1}-a^n>0 or a^{n-1}(1-a)>0.
If a>1, doesn't matter if n is even or odd, the given inequality cannot hold.
If a<0, then necessarily n must be odd.
Sufficient.

(2)a^n>a^{3n} can be rewritten as a^n-a^{3n} or a^n(1-a^{2n})>0.
Since a cannot be neither 0 nor 1, \,\,1-a^{2n} is certainly negative. (Now a cannot be -1 either.)
Then a^n is negative if a is negative and n is odd, and this is the only way the given inequality can hold.
If a is positive, doesn't matter the value of n, the inequality cannot hold.
Sufficient.

Answer D.
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Consider substituting numbers:

statement 1 : To get stmt 1 true n=2 (even) and a = -2 (+ve numbers will never satisfy condition)

-2^1>-2^2 = -2>4 (No)

n=3 (odd) and a = -2

-2^2>-2^3 = 4>-8 (Yes)

Sufficient

Statement 2 : To get stmt 2 true n=2 (even) and a = -2 (+ve numbers will never satisfy condition)

-2^2>-2^6 = 4>64 (No)

n=3 (odd) and a = -2

-2^3>-2^9 = -8>-512 (Yes)

Sufficient

Ans D

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Hi Bunuel....

I have read the explanation provided by Ian ,... but I'm unable to understand the explanation for 2nd statement ,, & I think Ian is no longer a regular visitor to the GMAT Club ..... Can You pls... Make me understand this question ....& solution Pls,,,,,, Thanks !!
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If a and n are integers and n > 1, is n odd?

(1) a^(n–1) > a^n
(2) a^n > a^(3n)

Well , i feel a bit intimidated by equation questions so it took me 2:25 to solve this one. But if you keep the initial hesitation aside , this can be solved in less than 2 min.

From 1) a^n-1 > a ^n. From given conditions ie n> 1 and a and n being integers , this is only possible if a is negative and n-1 is even integer. so n is an odd integer.
From 2) a^n > a^3n. Again only possible if a is negative and n has to be odd.

So D is the answer.

Can someone please explain me step-wise how the factorization is done?

Thanks.

If we start at the "end" I can see how it is the same, however..

=a^{n-1}-a^{n+1} ?

I don't see how these two share the ^(n-1)..

It's clear now. Thanks.