document.write( "Question 1135338: The sum of three numbers in G.P is 21 and the sum of their squares is 189.Find the numbers
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Algebra.Com's Answer #752921 by MathLover1(20850)\"\" \"About 
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\n" ); document.write( "In General we write a Geometric Sequence like this:\r
\n" ); document.write( "\n" ); document.write( "{ \"a\", \"ar\", \"ar%5E2\", \"ar%5E3\", ... } where:\r
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\n" ); document.write( "\n" ); document.write( "\"a\" is the first term, and
\n" ); document.write( "\"r\" is the factor between the terms (called the \"common ratio\")\r
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\n" ); document.write( "\n" ); document.write( "if the sum of three numbers in G.P is \"21\", we have\r
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\n" ); document.write( "\n" ); document.write( " \"a%2Bar%2Bar%5E2=21\"....solve for \"a\"\r
\n" ); document.write( "\n" ); document.write( "\"a%281%2Br%2Br%5E2%29=21\"\r
\n" ); document.write( "\n" ); document.write( "\"a=21%2F%281%2Br%2Br%5E2%29\".....eq.1\r
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\n" ); document.write( "\n" ); document.write( " and, if the sum of their squares is \"189\", we have\r
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\n" ); document.write( "\n" ); document.write( " \"a%5E2%2B%28ar%29%5E2%2B%28ar%5E2%29%5E2=189\"...solve for \"a\"\r
\n" ); document.write( "\n" ); document.write( " \"a%5E2%2Ba%5E2+r%5E2%2Ba%5E2r%5E4=189\"\r
\n" ); document.write( "\n" ); document.write( "\"a%5E2%281%2Br%5E2%2Br%5E4%29=189\"\r
\n" ); document.write( "\n" ); document.write( "\"a%5E2=189%2F%281%2Br%5E2%2Br%5E4%29\"\r
\n" ); document.write( "\n" ); document.write( "\"a=sqrt%28189%2F%281%2Br%5E2%2Br%5E4%29%29\"........eq.2\r
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\n" ); document.write( "\n" ); document.write( "from eq.1 and eq.2 we have\r
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\n" ); document.write( "\n" ); document.write( "\"21%2F%281%2Br%2Br%5E2%29=sqrt%28189%2F%281%2Br%5E2%2Br%5E4%29%29\"........square both sides\r
\n" ); document.write( "\n" ); document.write( "\"441%2F%281%2Br%2Br%5E2%29%5E2=189%2F%281%2Br%5E2%2Br%5E4%29\"\r
\n" ); document.write( "\n" ); document.write( "\"441%281%2Br%5E2%2Br%5E4%29=189%281%2Br%2Br%5E2%29%5E2\"\r
\n" ); document.write( "\n" ); document.write( "\"441%281%2Br%5E2%2Br%5E4%29=189%28r%5E4+%2B+2+r%5E3+%2B+3+r%5E2+%2B+2+r+%2B+1%29\"\r
\n" ); document.write( "\n" ); document.write( "\"441%2B441r%5E2%2B441r%5E4=189+r%5E4+%2B+378+r%5E3+%2B+567+r%5E2+%2B+378+r+%2B+189\"\r
\n" ); document.write( "\n" ); document.write( "\"441%2B441r%5E2%2B441r%5E4-189+r%5E4+-+378+r%5E3+-+567+r%5E2+-+378+r+-+189=0\"\r
\n" ); document.write( "\n" ); document.write( "\"252+r%5E4+-+378+r%5E3+-+126+r%5E2+-+378+r+%2B+252=0\".........factor\r
\n" ); document.write( "\n" ); document.write( "\"126+%282+r+-+1%29+%28r+-+2%29+%28r%5E2+%2B+r+%2B+1%29=0\"\r
\n" ); document.write( "\n" ); document.write( "solutions:\r
\n" ); document.write( "\n" ); document.write( "if \"2r-1=0\"=>\"r=1%2F2\"\r
\n" ); document.write( "\n" ); document.write( "if \"r-2=0\"=>\"r=2\"\r
\n" ); document.write( "\n" ); document.write( "if \"%28r%5E2+%2B+r+%2B+1%29=0\"=> use quadratic formula and you will get complex solutions
\n" ); document.write( "\"r+=+-%28-1%29%5E%281%2F3%29\"->disregard\r
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\n" ); document.write( "\n" ); document.write( "now find \"a\":\r
\n" ); document.write( "\n" ); document.write( "\"a=21%2F%281%2Br%2Br%5E2%29\".....eq.1, plug in \"r=1%2F2\"\r
\n" ); document.write( "\n" ); document.write( "\"a=21%2F%281%2B1%2F2%2B%281%2F2%29%5E2%29\"\r
\n" ); document.write( "\n" ); document.write( "\"a=21%2F%281%2B1%2F2%2B1%2F4%29\"\r
\n" ); document.write( "\n" ); document.write( "\"a=21%2F%284%2F4%2B2%2F4%2B1%2F4%29\"\r
\n" ); document.write( "\n" ); document.write( "\"a=21%2F%287%2F4%29\"\r
\n" ); document.write( "\n" ); document.write( "\"a=%2821%2A4%29%2F7\"\r
\n" ); document.write( "\n" ); document.write( "\"a=3%2A4\"\r
\n" ); document.write( "\n" ); document.write( "\"a=12\"\r
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\n" ); document.write( "\n" ); document.write( "or\r
\n" ); document.write( "\n" ); document.write( "\"a=21%2F%281%2Br%2Br%5E2%29\".....eq.1, plug in \"r=2\"\r
\n" ); document.write( "\n" ); document.write( "\"a=21%2F%281%2B2%2B2%5E2%29\"\r
\n" ); document.write( "\n" ); document.write( "\"a=21%2F7\"\r
\n" ); document.write( "\n" ); document.write( "\"a=3\"\r
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\n" ); document.write( "\n" ); document.write( "then \"a\", \"ar\", \"ar%5E2\" will be\r
\n" ); document.write( "\n" ); document.write( "if \"a=12\" and \"r=1%2F2\"\r
\n" ); document.write( "\n" ); document.write( "\"a=12\"\r
\n" ); document.write( "\n" ); document.write( "\"ar=12%2A%281%2F2%29=6\"\r
\n" ); document.write( "\n" ); document.write( " \"ar%5E2=12%2A%281%2F4%29=3\"\r
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\n" ); document.write( "\n" ); document.write( "or\r
\n" ); document.write( "\n" ); document.write( "if \"a=3\" and \"r=2\"\r
\n" ); document.write( "\n" ); document.write( "\"a=3\"\r
\n" ); document.write( "\n" ); document.write( "\"ar=3%2A2=6\"\r
\n" ); document.write( "\n" ); document.write( " \"ar%5E2=3%2A4=12\"\r
\n" ); document.write( "\n" ); document.write( "you actually got same terms in reverse order\r
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\n" ); document.write( "\n" ); document.write( "check their sum: \r
\n" ); document.write( "\n" ); document.write( "\"12%2B6%2B3=21\" and \r
\n" ); document.write( "\n" ); document.write( "\"12%5E2%2B6%5E2%2B3%5E2=189\" \r
\n" ); document.write( "\n" ); document.write( "\"144%2B36%2B9=189\" \r
\n" ); document.write( "\n" ); document.write( "\"189=189\" \r
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