document.write( "Question 1171708: The positive integers are arranged in the pattern illustrated below. If this pattern continues indefinitely, what is the number immediately above 39863\r
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Algebra.Com's Answer #797225 by Edwin McCravy(20060)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "row\r\n" );
document.write( " 1      1  2  5 10 17 26 ...   \r\n" );
document.write( " 2      3  4  7 12 19 28 ...\r\n" );
document.write( " 3      6  8  9 14 21 30 ...\r\n" );
document.write( " 4     11 13 15 16 23 32 ...\r\n" );
document.write( " 5     18 20 22 24 25 34 ...\r\n" );
document.write( " 6     27 29 31 33 35 36 ...\r\n" );
document.write( "...            ...                    ... \r\n" );
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document.write( "The diagonal elements are the squares, 1,4,9,16,25,... and the numbers right\r\n" );
document.write( "after the squares are 2,7,14,23,34, which means we have added 1,3,5,7,9... to\r\n" );
document.write( "each square to get the next term.  So the next term after the square n² is\r\n" );
document.write( " n² + the nth odd integer.  The nth odd integer is 2n-1.\r\n" );
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document.write( "Let's see if that's enough information to determine whether 39863 is a number\r\n" );
document.write( "to the right of a perfect square diagonal element or to the left of a perfect\r\n" );
document.write( "square diagonal element.\r\n" );
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document.write( "The square root of 39863 is 199.6572062. So either\r\n" );
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document.write( "39863 is on the 199th row to the right of diagonal element 199²=39601\r\n" );
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document.write( "OR\r\n" );
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document.write( "39863 is on the 200th row to the left of the diagonal element 200²=40000.\r\n" );
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document.write( "The term just right of 199² is 199² plus the 199th positive odd number, which\r\n" );
document.write( "is 2(199)-1=397.  But when we add 397 to the diagonal element 199²=39601, we\r\n" );
document.write( "get 39998 which is more than 39863, so 39863 cannot be to the right of 39601,\r\n" );
document.write( "so 39863 must be on the 200th row to the left of the diagonal element\r\n" );
document.write( "200²=40000.\r\n" );
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document.write( "The number just left of each square on the diagonal is 1 less than the square\r\n" );
document.write( "on the diagonal.  \r\n" );
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document.write( "Now let's observe the numbers left of the diagonal.\r\n" );
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document.write( "Let's look at the numbers left of the diagonal elements beginning with the 2nd\r\n" );
document.write( "row, as there are no numbers left of the 1 on the first row. They are\r\n" );
document.write( "3,8,15,24,..., which are 1 less than the squares on the diagonal. We observe\r\n" );
document.write( "they are the last terms of an arithmetic sequence with common difference 2.\r\n" );
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document.write( "So the 200th term on the 200th row is 40000 \r\n" );
document.write( "The 199th term is 1 less or 39999.\r\n" );
document.write( "The common ratio is 2.\r\n" );
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document.write( "\"a%5Bn%5D=a%5B1%5D%2B%28n-1%29d\"\r\n" );
document.write( "\"a%5B199%5D=a%5B1%5D%2B%28199-1%29%282%29\"\r\n" );
document.write( "\"39999=a%5B1%5D%2B%28198%29%282%29\"\r\n" );
document.write( "\"39999=a%5B1%5D%2B396\"\r\n" );
document.write( "\"39603=a%5B1%5D\"   <--- the first number on the 200th row\r\n" );
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document.write( "Now we find the number of term on the 200th row which 39863 is.\r\n" );
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document.write( "\"39863=39603%2B%28n-1%29%282%29\"\r\n" );
document.write( "\"260=2n-2\"\r\n" );
document.write( "\"262=2n\"\r\n" );
document.write( "\"131=n\"\r\n" );
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document.write( "So 39863 is the 131st term on the 200th row.\r\n" );
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document.write( "So we must now find the number immediately above that, which is\r\n" );
document.write( "the 131st term on the 199th row.\r\n" );
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document.write( "the 199th term of the 199th row is 39601 \r\n" );
document.write( "The 198th term is 1 less or 39600.\r\n" );
document.write( "The common ratio is 2.\r\n" );
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document.write( "\"a%5Bn%5D=a%5B1%5D%2B%28n-1%29d\"\r\n" );
document.write( "\"a%5B198%5D=a%5B1%5D%2B%28198-1%29%282%29\"\r\n" );
document.write( "\"39600=a%5B1%5D%2B%28197%29%282%29\"\r\n" );
document.write( "\"39600=a%5B1%5D%2B394\"\r\n" );
document.write( "\"39206=a%5B1%5D\"   <--- the first number on the 199th row\r\n" );
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document.write( "Now we find the 131st term on the 199th row\r\n" );
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document.write( "\"a%5B131%5D=39206%2B%28131-1%29%282%29\"\r\n" );
document.write( "\"a%5B131%5D=39206%2B%28130%29%282%29\"\r\n" );
document.write( "\"a%5B131%5D=39466\"\r\n" );
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document.write( "So 39466 is the 131st term on the 199th row, which is the term immediately\r\n" );
document.write( "above 39863.\r\n" );
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document.write( "Answer = 39466\r\n" );
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document.write( "Edwin
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