Difference between revisions of "1990 AIME Problems/Problem 1"
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== Problem == | == Problem == | ||
| − | The increasing sequence <math>2,3,5,6,7,10,11,\ldots</math> consists of all positive | + | The [[increasing sequence]] <math>2,3,5,6,7,10,11,\ldots</math> consists of all [[positive integer]]s that are neither the [[perfect square | square]] nor the [[perfect cube | cube]] of a positive integer. Find the 500th term of this sequence. |
== Solution == | == Solution == | ||
| − | + | Because there aren't that many perfect squares or cubes, let's look for the smallest perfect square greater than <math>500</math>. This happens to be <math>23^2=529</math>. Notice that there are <math>23</math> squares and <math>8</math> cubes less than or equal to <math>529</math>, but <math>1</math> and <math>2^6</math> are both squares and cubes. Thus, there are <math>529-23-8+2=500</math> numbers in our sequence less than <math>529</math>. Magically, we want the <math>500th</math> term, so our answer is the smallest non-square and non-cube less than <math>529</math>, which is <math>528</math>. | |
| − | Because there aren't that many perfect squares or cubes, let's look for the smallest perfect square greater than <math>500</math>. This happens to be <math>23^2=529</math>. Notice that there are <math>23</math> squares and <math>8</math> cubes less than or equal to <math>529</math>, but <math>1</math> and <math>2^6</math> are both squares and cubes. Thus, there are <math>529-23-8+2=500</math> numbers in our sequence less than <math>529</math>. Magically, we want the <math>500th</math> term, so our answer is the smallest non-square and non-cube less than <math>529</math>, which is <math>528</math>. | ||
== See also == | == See also == | ||
* [[1990 AIME Problems]] | * [[1990 AIME Problems]] | ||
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| + | [[Category:Intermediate Algebra Problems]] | ||
Revision as of 21:55, 25 February 2007
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 
.
