document.write( "Question 1210284: The sum of three numbers in a geometric progression GP is 13 and there products is -64. Find the numbers. \n" ); document.write( "

Algebra.Com's Answer #851850 by ikleyn(52781) $\"\"$ $\"About$

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\n" ); document.write( "The sum of three $\"highlight%28consecutive%29\"$ numbers in a geometric progression $\"highlight%28cross%28GP%29%29\"$ is 13 and $\"highlight%28cross%28there%29%29\"$ their $\"highlight%28cross%28products%29%29\"$ product is -64.
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document.write( "So, in standard designations, the terms are a, ar and ar^2.\r\n" );
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document.write( "From the problem, we have two equations\r\n" );
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document.write( "    a + ar + ar^2 =  13,     (1)   (the sum)\r\n" );
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document.write( "    a*(ar)*(ar^2) = -64.     (2)   (the product).\r\n" );
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document.write( "Equation (2) is the same as\r\n" );
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document.write( "    (ar)^3 = -64,\r\n" );
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document.write( "which gives us\r\n" );
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document.write( "    ar = -4.     (3)\r\n" );
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document.write( "We substitute ar = -4 into equation (1).  We get then\r\n" );
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document.write( "    a - 4 + ar^2 = 13,\r\n" );
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document.write( "    a + ar^2 = 13 + 4 = 17.    (4)\r\n" );
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document.write( "We also represent ar^2 as  (ar)*r = -4r.\r\n" );
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document.write( "Then equation (4) takes the form\r\n" );
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document.write( "    a - 4r = 17.    (5)\r\n" );
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document.write( "Now we have a system of two equations (3) and (5)\r\n" );
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document.write( "    ar = -4,        (3)\r\n" );
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document.write( "    a - 4r = 17.    (5)\r\n" );
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document.write( "    \r\n" );
document.write( "From (3), express r =   and substitute it into (5).  You will get\r\n" );
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document.write( "     = 17,\r\n" );
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document.write( "     a^2 + 16 = 17a,\r\n" );
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document.write( "     a^2 - 17a + 16 = 0,\r\n" );
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document.write( "     (a-16)*(a-1) = 0.\r\n" );
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document.write( "It gives two possible solutions for 'a', 16 and 1.\r\n" );
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document.write( "    If a = 16,  then  r =  =  = .\r\n" );
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document.write( "    If a =  1,  then  r =  =  =  -4.\r\n" );
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document.write( "Thus, two progressions are possible.\r\n" );
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document.write( "One progression is        16, -4, 1.                (*)\r\n" );
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document.write( "The other progression is   1, -4, 16   (the same as (*), but reversed)\r\n" );
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document.write( "Both progressions satisfy the imposed conditions.\r\n" );
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