document.write( "Question 838228: What is the sequence formula for 20, 4, 8, 16,13 \n" ); document.write( "
Algebra.Com's Answer #506153 by Edwin McCravy(20059) $\"\"$ $\"About$
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document.write( "To get the next term, there are two cases:\r\n" );
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document.write( "Case 1:  If a term has just 1 digit other than 0, the next term is\r\n" );
document.write( "twice that non-zero digit.\r\n" );
document.write( "Case 2:  If a term has no 0 digits, the next digit is the sum of the\r\n" );
document.write( "product and the sum of the digits. \r\n" );
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document.write( "[This defines all terms. All terms will be 1 or 2 digit numbers.  For\r\n" );
document.write( "even if a term were 99, case 2 would make the next term also 99.]\r\n" );
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document.write( "We start with 26.  26 is the 1st term.\r\n" );
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document.write( "26 is case 2.  The product of the digits is \r\n" );
document.write( "2×6 or 12. The sum of the digits is 2+6 or 8.  So the 2nd term is 12+8 or 20.\r\n" );
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document.write( "20 is case 1.  Twice its non-zero digit is 4.  So the 3rd term is 4.\r\n" );
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document.write( "4 is case 1.  Twice 4 is 8.  So the 4rd term is 8.\r\n" );
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document.write( "8 is case 1.  Twice 8 is 16.  So the 5th term is 16.\r\n" );
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document.write( "16 is case 2.  The product of the digits is 1×6 or 6. The sum of the digits \r\n" );
document.write( "is 1+6 or 7. So the 6th term is 6+7 or 13.\r\n" );
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document.write( "13 is case 2.  The product of the digits is 1×3 or 3. The sum of the digits \r\n" );
document.write( "is 1+3 or 4. So the 7th term is 3+4 or 7.\r\n" );
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document.write( "7 is case 1.  Twice 7 is 14.  So the 8th term is 14.\r\n" );
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document.write( "14 is case 2.  The product of the digits is 1×4 or 4. The sum of the digits \r\n" );
document.write( "is 1+4 or 5.  So the 9th term is 4+5 or 9.\r\n" );
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document.write( "9 is case 1.  Twice 9 is 18.  So the 10th term is 18.\r\n" );
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document.write( "18 is case 2.  The product of the digits is 1×8 or 8. The sum of the digits \r\n" );
document.write( "is 1+8 or 9. So the 11th term is 8+9 or 17.\r\n" );
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document.write( "17 is case 2.  The product of the digits is 1×7 or 7. The sum of the digits \r\n" );
document.write( "is 1+7 or 8.  So the 12th term is 7+8 or 15.\r\n" );
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document.write( "15 is case 2.  The product of the digits is 1×5 or 5. The sum of the digits \r\n" );
document.write( "is 1+5 or 6.  So the 13th term is 5+6 or 11.\r\n" );
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document.write( "11 is case 2.  The product of the digits is 1×1 or 1. The sum of the digits \r\n" );
document.write( "is 1+1 or 2.  So the 14th term is 1+2 or 3.\r\n" );
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document.write( "3 is case 1.  Twice 3 is 6.  So the 15th term is 6.\r\n" );
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document.write( "6 is case 1.  Twice 6 is 12.  So the 16th term is 12.\r\n" );
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document.write( "12 is case 2.  The product of the digits is 1×2 or 2. The sum of the digits \r\n" );
document.write( "is 1+2 or 3.  So the 17th term is 2+3 or 5.\r\n" );
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document.write( "5 is case 1.  Twice 5 is 10.  So the 18th term is 10.\r\n" );
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document.write( "10 is case 1.  Twice its non-zero digit is 2.  So the 19th term is 2.\r\n" );
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document.write( "2 is case 1.  Twice 2 is 4.  So the 20th term is 4.\r\n" );
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document.write( "Notice that 4 was also the 3rd term.  Therefore the sequence will\r\n" );
document.write( "repeat the 3rd through 19th terms infinitely:\r\n" );
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document.write( "26, 20, 4, 8, 16, 13, 7, 14, 9, 18, 17, 15, 11, 3, 6, 12, 5, 10, 2,\r\n" );
document.write( "        4, 8, 16, 13, 7, 14, 9, 18, 17, 15, 11, 3, 6, 12, 5, 10, 2,\r\n" );
document.write( "        4, 8, 16, 13, 7, 14, 9, 18, 17, 15, 11, 3, 6, 12, 5, 10, 2,...\r\n" );
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document.write( "That defines all the terms.\r\n" );
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document.write( "There is no easy general term for a sequence \r\n" );
document.write( "when each term of the sequence depends on the \r\n" );
document.write( "size and the base 10 digits of the preceding\r\n" );
document.write( "term as well as whether the preceding term\r\n" );
document.write( "ended in 0.  Here is the best thing you can do:\r\n" );
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document.write( "Define only the first 19 terms by writing them out\r\n" );
document.write( "according to the above rules about digits and size.\r\n" );
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document.write( "a(1)=26, a(2)=20, a(3)=4, a(4)=8, a(5)=16, \r\n" );
document.write( "a(6)=13, a(7)=7, a(8)=14, a(9)=9, a(10)=18,\r\n" );
document.write( "a(11)=17, a(12)=15, a(13)=11, a(14)=3, \r\n" );
document.write( "a(15)=6, a(16)=12, a(17)=5, a(18)=10,\r\n" );
document.write( "a(19)=2\r\n" );
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document.write( "Then for the nth term, give these two formulas \r\n" );
document.write( "when n > 19\r\n" );
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document.write( "a(n) = a(r) where r = the remainder when \r\n" );
document.write( "n is divided by 17 and r > 2\r\n" );
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document.write( "a(n) = a(r+17) where r = the remainder \r\n" );
document.write( "when n is divided by 17 and r < 3  \r\n" );
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document.write( "For instance, \r\n" );
document.write( "if you want to find a(932),\r\n" );
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document.write( "           54\r\n" );
document.write( "Divide 17)932\r\n" );
document.write( "          85\r\n" );
document.write( "           82\r\n" );
document.write( "           68\r\n" );
document.write( "           14\r\n" );
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document.write( "The remainder is r=14, so r > 2, and \r\n" );
document.write( "so a(932) = a(14) = 3 \r\n" );
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document.write( "if you want to find a(306), \r\n" );
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document.write( "           18\r\n" );
document.write( "Divide 17)306\r\n" );
document.write( "          17\r\n" );
document.write( "          136\r\n" );
document.write( "          136\r\n" );
document.write( "            0\r\n" );
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document.write( "The remainder r=0, so r < 2, and\r\n" );
document.write( "so a(r+17) = a(0+17)=a(17)=5 \r\n" );
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document.write( "From that you can always get the nth term.\r\n" );
document.write( "There is no easier formula for a(n), as this\r\n" );
document.write( "is an extremely complicated sequence.\r\n" );
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document.write( "Edwin
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