Question 1161108: The sequence 2, 3, 5, 6, 7, 10, ... consists of all counting numbers which are neither perfect squares nor perfect cubes. Find the 75th term of this sequence.
Found 2 solutions by Alan3354, ikleyn:
Answer by Alan3354(69443)   (Show Source):
You can put this solution on YOUR website! The sequence 2, 3, 5, 6, 7, 10, ... consists of all counting numbers which are neither perfect squares nor perfect cubes. Find the 75th term of this sequence.
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I don't think there's an algebraic solution to this.
Just a matter of grinding it out.
Also, I don't think there's much to be learned from it. Waste of time.
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Good luck.
Answer by ikleyn(52786)  (Show Source):
You can put this solution on YOUR website! .
Write all consecutive integer numbers 1, 2, 3, 4, 5 and so on, until, let say, 100.
From this sequence, erase perfect cubes and perfect squares
1, 4, 8, 9, 16, 25, 27, 36, 49, 64, 81, 100.
In all, you will erase 12 terms; 11 of them are less than or equal to 81.
Now you can easily understand that the 75-th term of your sequence is the number 86,
+----------------------------------------------------------------------------+
| because 86 original terms MINUS 11 erased terms = 75 remaining terms. |
+----------------------------------------------------------------------------+
ANSWER. The 75-th term of your sequence is the number 86.
Solved.
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It is a good problem for 4-th - 5-th grade kids, of the Math circle or Math Olympiad level
at a school or local University.
First, it teaches to think;
second, it teaches on how to find an idea;
third, it teaches to organize your thoughts;
fourth, it teaches to EXPRESS your thoughts;
fifth, it teaches that even Math problem, which seems to be tedious, can be solved and made as sweet as a cookie (!)
Come again to the forum soon to learn something new from me (!)
Bring nice problems with you (!)
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