If n is an integer and n2 is a positive integer, which must also be positive?
a. n2 + n
b. n2 – m3
c. 2n2 – n
d. n3 + n
e. 2n3 + 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, n3 will be greater than or equal to n2. So when subtracting it from n2 you will either get 0, if n is 1 or negative if n >1.
For answer a, if n = -1, then n2 + 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 2n2 – n > 0 for any positive number n.
If n is negative, 2n2 is positive and subtracting a negative is like adding a positive, so 2n2 – n > 0 for any negative number n as well.