document.write( "Question 1120323: For a geometric series, S2 = 20 and S3= 65. Find the first 3 terms. \n" ); document.write( "

Algebra.Com's Answer #735995 by ikleyn(52790) $\"\"$ $\"About$

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document.write( "From the condition, we have\r\n" );
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document.write( " =  =  = a*(1+r),              (1)\r\n" );
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document.write( " =  =  = .    (2)\r\n" );
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document.write( "Divide (1) by (2). You will get\r\n" );
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document.write( " =  =  = .\r\n" );
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document.write( "or\r\n" );
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document.write( "13(1+r) = 4*(1+r+r^2),\r\n" );
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document.write( "13 + 13r = 4 + 4r + 4r^2\r\n" );
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document.write( "4r^2 -9r - 9 = 0\r\n" );
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document.write( " =  = ;\r\n" );
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document.write( " =  = 3;    =  =  = .\r\n" );
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document.write( "Subtract (1) from (2). You will get \r\n" );
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document.write( " =  -  = 65 - 20 = 45 = .     (*)\r\n" );
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document.write( "Thus we should consider TWO opportunities:\r\n" );
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document.write( "    1.  If r = 3,  then according to (*)   = 45 =  = 9a,\r\n" );
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document.write( "        which implies  a = 5.\r\n" );
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document.write( "        Then   = 5,   = 3*5 = 15  and   = 3*15 = 45.\r\n" );
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document.write( "        Then   =  = 5 + 15 = 20  and   = 5 + 15 + 45 = 65    ! Correct !.\r\n" );
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document.write( "    2.  If r= , then according (*)  a =  = 5*16 = 80.  \r\n" );
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document.write( "        Then   =  = -60  and   =  = 45.\r\n" );
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document.write( "        In this way,   =  = 80 - 60 = 20  \r\n" );
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document.write( "                and    =  = 80 - 60 + 45 = 65.\r\n" );
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document.write( "        So, this opportunity does work, too.\r\n" );
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document.write( "Answer.  There are  TWO solutions.\r\n" );
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document.write( "         One solution is  a= 5, r= 3 and the three terms are  5, 15, 45.\r\n" );
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document.write( "         Another solution is  a= 80,  r=   and the three terms are  80, -60 and 45.\r\n" );
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