1990 AIME Problems/Problem 1

Problem

The increasing sequence $2,3,5,6,7,10,11,\ldots$ consists of all positive integers that are neither the square nor the cube of a positive integer. Find the 500th term of this sequence.

Solution

Because there aren't that many perfect squares or cubes, let's look for the smallest perfect square greater than $500$ . This happens to be $23^2=529$ . Notice that there are $23$ squares and $8$ cubes less than or equal to $529$ , but $1$ and $2^6$ are both squares and cubes. Thus, there are $529-23-8+2=500$ numbers in our sequence less than $529$ . Magically, we want the $500th$ term, so our answer is the smallest non-square and non-cube less than $529$ , which is $\boxed{528}$ . $$ (Error compiling LaTeX. Unknown error_msg)

1990 AIME (Problems • Answer Key • Resources)
Preceded by First Question	Followed by Problem 2
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1990 AIME Problems/Problem 1

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