1990 AIME Problems/Problem 1
Problem
The increasing sequence 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 . This happens to be
. Notice that there are
squares and
cubes less than or equal to
, but
and
are both squares and cubes. Thus, there are
numbers in our sequence less than
. Magically, we want the
term, so our answer is the smallest non-square and non-cube less than
, which is
.