document.write( "Question 460555: The average of the squares of seven consecutives integers is 53. The average of these integers is? \n" ); document.write( "
Algebra.Com's Answer #315898 by Edwin McCravy(20055)\"\" \"About 
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The average of the squares of seven consecutives integers is 53. The average of these integers is?
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document.write( "The tutor above, who solved it before me, suggested doing it a\r\n" );
document.write( "harder way, by using\r\n" );
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document.write( "x,  x+1, ... , x+6.\r\n" );
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document.write( "But it's much easier to let x be the middle integer instead of\r\n" );
document.write( "the smallest integer.  I see that the tutor below, who solved it\r\n" );
document.write( "after I did, solved it the same way I did.\r\n" );
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document.write( "There are two solutions:  7 and -7.  Here's how to \r\n" );
document.write( "find them:\r\n" );
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document.write( "Let the 4th integer be x, then the 7 integers are\r\n" );
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document.write( "x-3, x-2, x-1, x, x+1, x+2, x+3\r\n" );
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document.write( "and their average is their sum over 7 \r\n" );
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document.write( "x-3+x-2+x-1+x+x+1+x+2+x+3\r\n" );
document.write( "覧覧覧覧覧覧覧覧覧覧覧覧予r\n" );
document.write( "            7\r\n" );
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document.write( "Collecting terms:\r\n" );
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document.write( "          7x\r\n" );
document.write( "          覧\r\n" );
document.write( "           7\r\n" );
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document.write( "           x\r\n" );
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document.write( "Their squares are:\r\n" );
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document.write( "(x-3)ｲ, (x-2)ｲ, (x-1)ｲ, xｲ, (x+1)ｲ, (x+2)ｲ, (x+3)ｲ\r\n" );
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document.write( "or\r\n" );
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document.write( "xｲ-6x+9, xｲ-4x+4, xｲ-2x+1, xｲ, xｲ+x+1, xｲ+4x+4, xｲ+6x+9\r\n" );
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document.write( "The average of these squares is the sum of those squares\r\n" );
document.write( "over 7\r\n" );
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document.write( "xｲ-6x+9+xｲ-4x+4+xｲ-2x+1+xｲ+xｲ+x+1+xｲ+4x+4+xｲ+6x+9\r\n" );
document.write( "覧覧覧覧覧覧覧覧覧覧覧覧覧覧覧覧覧覧覧覧覧覧覧覧予r\n" );
document.write( "                       7\r\n" );
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document.write( "Collecting terms:\r\n" );
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document.write( "7xｲ+28\r\n" );
document.write( "覧覧覧\r\n" );
document.write( "   7\r\n" );
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document.write( "Setting that equal to 53\r\n" );
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document.write( "7xｲ+28\r\n" );
document.write( "覧覧覧 = 53\r\n" );
document.write( "   7\r\n" );
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document.write( "Multiply through by 7\r\n" );
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document.write( "7xｲ+28 = 371\r\n" );
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document.write( "   7xｲ = 343\r\n" );
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document.write( "    xｲ = 49\r\n" );
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document.write( "     x = ｱ7\r\n" );
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document.write( "And since x is their average. the answer is ｱ7\r\n" );
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document.write( "So there are two solutions.\r\n" );
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document.write( "The integers are then either\r\n" );
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document.write( "-10, -9, -8, -7 , -6, -5, -4\r\n" );
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document.write( "           or\r\n" );
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document.write( "  4, 5, 6, 7, 8, 9, 10 \r\n" );
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document.write( "Checking: the average of the square of either are\r\n" );
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document.write( "  4ｲ + 5ｲ + 6ｲ + 7ｲ + 8ｲ + 9ｲ + 10ｲ\r\n" );
document.write( "  覧覧覧覧覧覧覧覧覧覧覧覧覧覧覧覧予r\n" );
document.write( "               7\r\n" );
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document.write( "  16 + 25 + 36 + 49 + 64 + 81 + 100\r\n" );
document.write( "  覧覧覧覧覧覧覧覧覧覧覧覧覧覧覧覧予r\n" );
document.write( "                 7\r\n" );
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document.write( "               371\r\n" );
document.write( "               覧予r\n" );
document.write( "                7\r\n" );
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document.write( "               53\r\n" );
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document.write( "So we're correct.  There are two solutions.\r\n" );
document.write( "Perhaps you were told only to find the positive \r\n" );
document.write( "consecutive integers. If not, give both solutions.\r\n" );
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document.write( "Edwin
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