document.write( "Question 1157937: The sum of the first three terms in a GP is 38. Their product is 1728. Find the values of the three terms.\r
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\n" ); document.write( "\n" ); document.write( "My answer were 8 and 18. I want to confirm if it's correct and how will I find the values of the three terms with two solution of a and r \n" ); document.write( "

Algebra.Com's Answer #780841 by ikleyn(52788) $\"\"$ $\"About$

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document.write( "Let \"a\" be the first term and \"r\" be the common ratio.\r\n" );
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document.write( "Then from the condition, we have these two equations\r\n" );
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document.write( "    a + ar + ar^2  =   38,      (1)\r\n" );
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document.write( "    a*(ar*)*(ar^2) = 1728.      (2)\r\n" );
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document.write( "From equation (2),  a^3*r^3 = 1728,  or  (ar)^3 = 1728,   which implies\r\n" );
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document.write( "    ar =  = 12;          (3)    \r\n" );
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document.write( "hence,  \r\n" );
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document.write( "    r  = .                   (4)\r\n" );
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document.write( "Now, in equation (1) replace the term  ar  by 12, based on (3).  You will get\r\n" );
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document.write( "    a + 12 + ar^2 =  38,   which implies\r\n" );
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document.write( "    a + ar^2 = 26.              (5)\r\n" );
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document.write( "Next, substitute  r =   into equation (5), replacing \"r\" there.  You will get\r\n" );
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document.write( "     +  = 26,   or\r\n" );
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document.write( "     +  = 26.\r\n" );
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document.write( "Multiply by \"a\" both sides and simplify\r\n" );
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document.write( "    a^2 - 26a + 144 = 0,\r\n" );
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document.write( "     - 169 + 144 = 0\r\n" );
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document.write( "     = 25\r\n" );
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document.write( "    a - 13 = +/-  = +/- 5.\r\n" );
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document.write( "Thus two solutions for \"a\" are  a = 13 + 5 = 18  or  a = 13 - 5 = 8.\r\n" );
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document.write( "If  a =  8, then from (4)  r =  = .\r\n" );
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document.write( "If  a = 18, then from (4)  r =  = .\r\n" );
document.write( "    \r\n" );
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document.write( "In the first case, if a = 8,  then the three terms are  8,  = 12  and   = 18.\r\n" );
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document.write( "    In this case, the sum of terms is  8 + 12 + 18 = 38, so this solution does work.\r\n" );
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document.write( "In the second case, if a = 18,  then the three terms are  18,  = 12  and   = 8.\r\n" );
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document.write( "    In this case, the sum of terms is  18 + 12 + 8 = 38, so this solution does work, too.\r\n" );
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document.write( "ANSWER.  The problem has two solution:  \r\n" );
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document.write( "         a)  first term is 18;  the common difference is   and the progression is  18, 12, 8.\r\n" );
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document.write( "         b)  first term is  8;  the common difference is   and the progression is   8, 12, 18.\r\n" );
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