document.write( "Question 1208497: 1/2, 2/3, 6/5,? \n" ); document.write( "
Algebra.Com's Answer #846941 by Edwin McCravy(20056)\"\" \"About 
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document.write( "Below are four possibilities, which is to show you there are more than\r\n" );
document.write( "one possible answer when given a partial sequence to find the next term.\r\n" );
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document.write( "Here is one possibility:\r\n" );
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document.write( "You start with a fraction with the smallest positive integer as its numerator, and the smallest prime integer as its denominator. \r\n" );
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document.write( "Then to get the next term, which is a fraction, multiply the numerator by the\r\n" );
document.write( "denominator to get the numerator of the next term, and use the next prime number\r\n" );
document.write( "as its denominator. \r\n" );
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document.write( "You start with 1/2 as the first term.\r\n" );
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document.write( "You multiply 1x2=2 to get the numerator of the 2nd term.\r\n" );
document.write( "You use the next prime 3 for a denominator. So the 2nd term is 2/3.\r\n" );
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document.write( "You multiply 2x3=6 to get the numerator of the 3rd term.\r\n" );
document.write( "You use the next prime 5 for a denominator. So the 3rd term is 6/5.\r\n" );
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document.write( "You multiply 6x5=30 to get the numerator of the 4th term.\r\n" );
document.write( "You use the next prime 7 for a denominator. So the 4th term is 30/7,\r\n" );
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document.write( "1/2, 2/3, 6/5, 30/7, ...\r\n" );
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document.write( "Here is another possibility:\r\n" );
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document.write( "The numerators are the factorials 1!=1, 2!=2, 3!=6, 4!=24\r\n" );
document.write( "The denominators are the positive integers which are not divisible by any\r\n" );
document.write( "perfect square other than 1. Neither 2, 3, 5, are divisible by a \r\n" );
document.write( "perfect square other than 1, so the next one that isn't is 7.\r\n" );
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document.write( "1/2, 2/3, 6/5, 24/7, ... \r\n" );
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document.write( "Here is another possibility:\r\n" );
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document.write( "The numerators of each odd-numbered term is 1 more than a multiple of 5.\r\n" );
document.write( "The numerators of each even-numbered term is 1 more than the numerator of the\r\n" );
document.write( "preceding odd-numbered term.\r\n" );
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document.write( "The denominator of each term is found by multiplying the number of term by one less, adding 4, then dividing by 2.\r\n" );
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document.write( "The 1st term is odd-numbered because 1 is an odd number. So the numerator is 1\r\n" );
document.write( "more than the 1st multiple of 5, which is 0. Add 1 and get 1. \r\n" );
document.write( "The denominator: 1x0=0, then add 4, get 4. Then divide by 2, get 2. \r\n" );
document.write( "So the 1st term is 1/2.\r\n" );
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document.write( "The 2nd term is even-numbered because 2 is an even number. So the numerator is 1\r\n" );
document.write( "more than the numerator of the preceding term, or 2 \r\n" );
document.write( "The denominator: 2x1=2, then add 4, get 6. Then divide by 2, get 3.  \r\n" );
document.write( "So the 2nd term is 2/3\r\n" );
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document.write( "The 3rd term is odd-numbered because 3 is an odd number. So the numerator is 1\r\n" );
document.write( "more than the 2nd multiple of 5, which is 5. Add 1 and get 6. \r\n" );
document.write( "The denominator: 3x2=6, then add 4, get 10. Then divide by 2, get 5. \r\n" );
document.write( "So the 1st term is 6/5.\r\n" );
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document.write( "The 4th term is even-numbered because 4 is an even number. So the numerator is 1\r\n" );
document.write( "more than the numerator of the preceding term, or 7 \r\n" );
document.write( "The denominator: 4x3=12, then add 4, get 16. Then divide by 2, get 8.  \r\n" );
document.write( "So the 4th term is 7/8.\r\n" );
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document.write( "1/2, 2/3, 6/5, 7/8, ...\r\n" );
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document.write( "Here is another possibility:\r\n" );
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document.write( "For the 1st term, n=1 \r\n" );
document.write( "(a) substitute n=1 in the quadratic 11n2-23n+42\r\n" );
document.write( "(b) get 11-23+42=30\r\n" );
document.write( "(c) divide by 60, get 30/60 which reduces to 1/3\r\n" );
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document.write( "For the 2nd term, n=2 \r\n" );
document.write( "(a) substitute n=2 in the quadratic 11n2-23n+42\r\n" );
document.write( "(b) get 44-46+42=40\r\n" );
document.write( "(c) divide by 60, get 40/60 which reduces to 2/3\r\n" );
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document.write( "For the 3rd term, n=3 \r\n" );
document.write( "(a) substitute n=3 in the quadratic 11n2-23n+42\r\n" );
document.write( "(b) get 99-69+42=72\r\n" );
document.write( "(c) divide by 60, get 72/60 which reduces to 6/5\r\n" );
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document.write( "For the 5th term, n=4 \r\n" );
document.write( "(a) substitute n=4 in the quadratic 11n2-23n+42\r\n" );
document.write( "(b) get 176-92+42=126\r\n" );
document.write( "(c) divide by 60, get 126/60 which reduces to 21/10\r\n" );
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document.write( "1/2, 2/3, 6/5, 21/10, ...\r\n" );
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document.write( "Edwin
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