document.write( "Question 1141725: Given 2,5,8 \r
\n" ); document.write( "\n" ); document.write( "Prove that none of the terms of this sequence are perfect squares
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Algebra.Com's Answer #762335 by greenestamps(13203)\"\" \"About 
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\n" ); document.write( "The problem as you pose it is trivial. \"This sequence\" is the three numbers 2, 5, and 8. None of 2, 5, or 8 is a perfect square.

\n" ); document.write( "The problem you intended to pose is an interesting one, involving the infinite arithmetic sequence 2, 5, 8, ....

\n" ); document.write( "The numbers in this sequence are the numbers of the form 3n+2, where n is an integer. We need to show that there are no squares of integers that are of that form.

\n" ); document.write( "Every integer can be represented in exactly one way as either 3k-1, 3k, or 3k+1, where k is an integer.

\n" ); document.write( "If the integer is of the form 3k-1 or 3k+1, then the square of the integer is of the form 3n+1:
\n" ); document.write( "\"%283k-1%29%5E2+=+9k%5E2-6k%2B1+=+3%283k%5E2-2k%29%2B1\"
\n" ); document.write( "\"%283k%2B1%29%5E2+=+9k%5E2%2B6k%2B1+=+3%283k%5E2%2B2k%29%2B1\"

\n" ); document.write( "And if the integer is of the form 3k, then the square of the integer is of the form 3n:
\n" ); document.write( "\"%283k%29%5E2+=+9k%5E2+=+3%283k%5E2%29\"

\n" ); document.write( "So the square of any integer is either of the form 3n or 3n+1 -- never 3n+2.
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