document.write( "Question 188687: will this pattern continue 39 x 41 = 40 x 40 - 1, 59 x 61 = 60 x 60 - 1 \n" ); document.write( "
Algebra.Com's Answer #141556 by Edwin McCravy(20066)\"\" \"About 
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document.write( "Write the pattern as A x B = C x C - 1\r\n" );
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document.write( "The pattern does appears to continue\r\n" );
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document.write( " (A x  B)  = (C x  C) - 1\r\n" );
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document.write( " 19 x  21 =  20 x  20 - 1     =   399\r\n" );
document.write( " 39 x  41 =  40 x  40 - 1     =  1599\r\n" );
document.write( " 59 x  61 =  60 x  60 - 1     =  3599\r\n" );
document.write( " 79 x  81 =  80 x  80 - 1     =  6399\r\n" );
document.write( " 99 x 101 = 100 x 100 - 1     =  9999\r\n" );
document.write( "119 x 121 = 120 x 120 - 1     = 14399\r\n" );
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document.write( "but we must find out for certain if the pattern\r\n" );
document.write( "continues forever:\r\n" );
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document.write( "The numbers in both C columns are multiples of 20,\r\n" );
document.write( "so the general expression for them is C = 20n\r\n" );
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document.write( "The numbers in column A are 1 less than the numbers \r\n" );
document.write( "in column C, so the general expression for them is\r\n" );
document.write( "A = 20n-1\r\n" );
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document.write( "The numbers in column B are 1 more than the numbers \r\n" );
document.write( "in column C, so the general expression for them is\r\n" );
document.write( "B = 20n+1\r\n" );
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document.write( "Now we want to find out if it is always true that'\r\n" );
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document.write( "        A x B  = C x C - 1\r\n" );
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document.write( "So we substitute \r\n" );
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document.write( "(20n-1)(20n+1) = (20n)(20n)-1 \r\n" );
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document.write( "Then we simplify each side:\r\n" );
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document.write( "       400n2-1 = 400n2-1 \r\n" );
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document.write( "so since both sides are the same, regardless\r\n" );
document.write( "of n, we now know for certain that the pattern \r\n" );
document.write( "will continue forever.\r\n" );
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document.write( "Edwin
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