document.write( "Question 471136: Prove that in an arithmetis sequence of natural numbers, one of the terms is aperfect square , then there are many terms which are perfect squares. Is there an arithmetic sequence of natural numbers in which no term is a perfect square? \n" ); document.write( "
Algebra.Com's Answer #323181 by Edwin McCravy(20056)\"\" \"About 
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Prove that if an arithmetis sequence of natural numbers, one of the terms is a perfect square, then there are many terms which are perfect squares.
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document.write( "Proof:\r\n" );
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document.write( "The nth term of an arithmetic sequence is\r\n" );
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document.write( "an = a1 + (n-1)d\r\n" );
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document.write( "Suppose this is a sequence of all natural numbers.\r\n" );
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document.write( "Then a1 and a2 are natural numbers\r\n" );
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document.write( "a2 = a1 + (2-1)d\r\n" );
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document.write( "a2-a1 = d\r\n" );
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document.write( "Therefore d is an integer.  d cannot be negative, otherwise\r\n" );
document.write( "there would eventally be negative terms in the sequence.\r\n" );
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document.write( "Case 1:\r\n" );
document.write( "d = 0.  Then the terms are all the same, and if any one\r\n" );
document.write( "of them is a perfect square, then they all are, and the \r\n" );
document.write( "theorem is proved.\r\n" );
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document.write( "Case 2:\r\n" );
document.write( "d is a natural number.\r\n" );
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document.write( "Suppose the kth term is a perfect square pē\r\n" );
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document.write( "ak = \r\n" );
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document.write( "           a1 + (k-1)d = pē\r\n" );
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document.write( "add 2pmd + mēdē to both sides where m is a natural number\r\n" );
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document.write( "a1 + (k-1)d + 2pmd + mēdē = pē + 2pmd + mēdē\r\n" );
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document.write( "Factor d out of the last three terms on the left\r\n" );
document.write( "and factor the right side as a perfect square:\r\n" );
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document.write( "     a1 + [(k-1)+2pm+mē]d = (p+md)ē\r\n" );
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document.write( "The right side is a perfect square, and the left side\r\n" );
document.write( "is the (k-1+2pm+mē)th term for infinitely many values of m.\r\n" );
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document.write( "Thus the theorem is proved.\r\n" );
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\n" ); document.write( "Is there an arithmetic sequence of natural numbers in which no term
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document.write( "A simple case is with d=0\r\n" );
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document.write( "2,2,2,2,2,2,2,...\r\n" );
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document.write( "But your teacher may not go for that as it is too trivial.\r\n" );
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document.write( "However this one doesn't contain any term that is a perfect square: \r\n" );
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document.write( "2,6,10,14,18,22,...\r\n" );
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document.write( "Here's why.  This sequence contains only even terms. If this \r\n" );
document.write( "sequence contained a term that was a perfect square, it would \r\n" );
document.write( "be even.  But every even perfect square is divisible by 4.\r\n" );
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document.write( "This sequence has a1=2, d=4, thus its nth term is: \r\n" );
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document.write( "an = a1+(n-1)d = 2+(n-1)*4 = 2+4n-4 = 4n-2.\r\n" );
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document.write( "But if we divide 4n-2 by 4 we get n-\"1%2F2\" which is not \r\n" );
document.write( "an integer.  So the sequence cannot contain a perfect square.\r\n" );
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document.write( "Edwin
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