document.write( "Question 683947: the first two terms of a sequence are 4 and 5, and each subsequent term is the sum of the two preceding terms. how many of the first 1000 terms are odd?
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Algebra.Com's Answer #423867 by Edwin McCravy(20055)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "We use the facts that\r\n" );
document.write( "An even number + an odd number = an odd number\r\n" );
document.write( "An odd number + an odd number = an even number\r\n" );
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document.write( "The 1st term is 4 which is even\r\n" );
document.write( "The 2nd term is 5 which is odd\r\n" );
document.write( "The 3rd term is an even + an odd which is odd\r\n" );
document.write( "The 4th term is an odd + an odd which is even\r\n" );
document.write( "The 5th term is an odd + an even which is odd\r\n" );
document.write( "The 5th term is an odd + an even which is odd\r\n" );
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document.write( "So the pattern is:\r\n" );
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document.write( " 1. even \r\n" );
document.write( " 2. odd \r\n" );
document.write( " 3. odd\r\n" );
document.write( " 4. even\r\n" );
document.write( " 5. odd\r\n" );
document.write( " 6. odd\r\n" );
document.write( " 7. even \r\n" );
document.write( " 8. odd\r\n" );
document.write( " 9. odd\r\n" );
document.write( "10. even\r\n" );
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document.write( "It will be easier to find the number of evens than the number of \r\n" );
document.write( "odds in that sequence.  Then we can subtract from 1000 to get the\r\n" );
document.write( "number of odds.\r\n" );
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document.write( "The sequence of term NUMBERS of evens in that sequence is\r\n" );
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document.write( "1,4,7,10,...\r\n" );
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document.write( "That is an arithmetic sequence with first term a1 = 1, \r\n" );
document.write( "and common difference d = 3\r\n" );
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document.write( "[Don't get confused here because we have a new ARITHMETIC sequence \r\n" );
document.write( "whose terms themselves are the term NUMBERS of evens in the original \r\n" );
document.write( "sequence, which is NOT an arithmetic sequence.]\r\n" );
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document.write( "an = a1 + (n-1)d\r\n" );
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document.write( "an = 1 + (n-1)(3)\r\n" );
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document.write( "an = 1 + 3(n-1)\r\n" );
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document.write( "an = 1 + 3n - 3\r\n" );
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document.write( "an = 3n - 2\r\n" );
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document.write( "There are 1000 terms in the original sequence. Therefore\r\n" );
document.write( "all term numbers of the original sequence are 1000 or less:\r\n" );
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document.write( "    an < 1000 \r\n" );
document.write( "3n - 2 < 1000\r\n" );
document.write( "    3n < 1002\r\n" );
document.write( "     n < 334\r\n" );
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document.write( "So term number a334 = 3(334) - 2 = 1002 - 2 = 1000\r\n" );
document.write( "That is, term number 1000, the last term in the original sequence\r\n" );
document.write( "is even, so there are 334 term NUMBERS in the original sequence that\r\n" );
document.write( "are term NUMBERS of evens.  So there are 334 terms in the original \r\n" );
document.write( "sequence that are even.\r\n" );
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document.write( "However we were asked for the number of odd terms in the\r\n" );
document.write( "original sequence, so the answer is 1000 - 334 = 666.\r\n" );
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document.write( "Answer: 666  [the mark of the beast! :)  ]\r\n" );
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document.write( "Edwin
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