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If n is a perfect square, then is n+2 not a perfect square?

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Final answer:

If n is a perfect square, n + 2 is generally not a perfect square, because the sum would fall between the squares of two consecutive integers.

Step-by-step explanation:

The question asks if n+2 is not a perfect square assuming n is a perfect square. Considering n to be a perfect square, we can represent it as n = x², where x is an integer. When we examine n + 2, that will be x² + 2. In general, the sum of a perfect square and 2 cannot be a perfect square because perfect squares are consecutive integers squared and adding 2 would place the value in between two consecutive squares, making it impossible to be a square itself. For example, the squares of consecutive integers 3 and 4 are 9 and 16 respectively, and 9 + 2 = 11, which is not a perfect square - it lies between 9 and 16.

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