document.write( "Question 1187213: The sum of four numbers in an arithmetic progression is 98. The sum of their squares is 3006. Find the third number. \n" ); document.write( "

Algebra.Com's Answer #818193 by ikleyn(52803) $\"\"$ $\"About$

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\n" ); document.write( "The sum of four numbers in an arithmetic progression is 98.
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document.write( "Let  ,  ,  ,    be four terms of the AP.\r\n" );
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document.write( "Let  \"c\"  be the central point in the number line, exactly half way between the terms  and .\r\n" );
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document.write( "Let d be the  HALF  of the common difference of the progression.\r\n" );
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document.write( "Then   = c - 3d,\r\n" );
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document.write( "       = c -  d,\r\n" );
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document.write( "       = c +  d,\r\n" );
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document.write( "       = c + 3d.\r\n" );
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document.write( "Then the sum of the four terms is 4d,  and it equals 98, so  c = 98/4 = 24.5\r\n" );
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document.write( "The sum of squares of the four terms is\r\n" );
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document.write( "    (c-3d)^2 + (c-d)^2 + (c+d)^2 + (c+3d)^2 = 3006\r\n" );
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document.write( "Making FOIL and combining like terms, you arrive to equation\r\n" );
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document.write( "    4c^2 + 20d^2 = 3006,\r\n" );
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document.write( "    4*24.5^2 + 20d^2 = 3006\r\n" );
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document.write( "    20d^2 = 3006 - 4*24.5^2 = 605\r\n" );
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document.write( "      d^2 = 605/20 = 30.25\r\n" );
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document.write( "      d =  = +/- 5.5.\r\n" );
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document.write( "Thus the four terms of the progression are  24.5-3*5.5 = 8,      24.5-5.5 = 19,     30,  41,  if  d= 5.5,  and\r\n" );
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document.write( "                                            24.5-3*(-5.5) = 41,  24.5-(-5.5) = 30,  19,   8,  if d= -5.5.\r\n" );
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document.write( "Thus the problem has two possible  ANSWERS  for the third term: it is EITHER  30  OR  19.\r\n" );
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