If n is an integer and n

^{2}is a positive integer, which must also be positive?
a. n

^{2}+ n
b. n

^{2}– m^{3}
c. 2n

^{2}– n
d. n

^{3}+ n
e. 2n

^{3}+ n
At
first glance a,b and c all seem reasonable because squaring any
number will give a positive answer. But answer b wont always work
because if n is positive, n

^{3}will be greater than or equal to n^{2}. So when subtracting it from n^{2}you will either get 0, if n is 1 or negative if n >1.
For
answer a, if n = -1, then n

^{2}+ n = 0, which does not give a positive answer either.
Therefore
answer c is the correct answer.

Any
positive number squared times 2 will be greater than the number
itself so 2n

^{2}– n > 0 for any positive number n.
If
n is negative, 2n

^{2}is positive and subtracting a negative is like adding a positive, so 2n^{2}– n > 0 for any negative number n as well.
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