1990 AIME Problems/Problem 1: Difference between revisions
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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. | 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 1== | ||
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 | 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 biggest non-square and non-cube less than <math>529</math>, which is <math>\boxed{528}</math>. | ||
== Solution 2== | |||
This solution is similar as Solution 1, but to get the intuition why we chose to consider <math>23^2 = 529</math>, consider this: | |||
We need <math>n - T = 500</math>, where <math>n</math> is an integer greater than 500 and <math>T</math> is the set of numbers which contains all <math>k^2,k^3\le 500</math>. | |||
Firstly, we clearly need <math>n > 500</math>, so we substitute n for the smallest square or cube greater than <math>500</math>. However, if we use <math>n=8^3=512</math>, the number of terms in <math>T</math> will exceed <math>n-500</math>. Therefore, <math>n=23^2=529</math>, and the number of terms in <math>T</math> is <math>23+8-2=29</math> by the [[Principle of Inclusion-Exclusion]], fulfilling our original requirement of <math>n-T=500</math>. | |||
As a result, our answer is <math>529-1 = \boxed{528}</math>. | |||
== See also == | == See also == | ||
{{AIME box|year=1990|before=First Question|num-a=2}} | {{AIME box|year=1990|before=First Question|num-a=2}} | ||
[[Category:Intermediate Algebra Problems]] | [[Category:Intermediate Algebra Problems]] | ||
{{MAA Notice}} | |||
Latest revision as of 22:11, 25 June 2023
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 1
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 biggest non-square and non-cube less than , which is 
.
Solution 2
This solution is similar as Solution 1, but to get the intuition why we chose to consider 
, consider this:
We need 
, where 
is an integer greater than 500 and 
is the set of numbers which contains all 
.
Firstly, we clearly need 
, so we substitute n for the smallest square or cube greater than . However, if we use 
, the number of terms in will exceed 
. Therefore, 
, and the number of terms in is 
by the Principle of Inclusion-Exclusion, fulfilling our original requirement of 
.
As a result, our answer is 
.
See also
| 1990 AIME (Problems • Answer Key • Resources) | ||
| Preceded by First Question |
Followed by Problem 2 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
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