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 1

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 biggest non-square and non-cube less than $529$ , which is $\boxed{528}$ .

Solution 2

This solution is similar as Solution 1, but to get the intuition why we chose to consider $23^2 = 529$ , consider this:

We need $n - T = 500$ , where $n$ is an integer greater than 500 and $T$ is the set of numbers which contains all $k^2,k^3\le 500$ .

Firstly, we clearly need $n > 500$ , so we substitute n for the smallest square or cube greater than $500$ . However, if we use $n=8^3=512$ , the number of terms in $T$ will exceed $n-500$ . Therefore, $n=23^2=529$ , and the number of terms in $T$ is $23+8-2=29$ by the Principle of Inclusion-Exclusion, fulfilling our original requirement of $n-T=500$ . As a result, our answer is $529-1 = \boxed{528}$ .

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

Contents

Problem

Solution 1

Solution 2

See also