document.write( "Question 1199359: In a geometric sequence of real numbers, the sum of the first
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Algebra.Com's Answer #833210 by ikleyn(52788) $\"\"$ $\"About$

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\n" ); document.write( "In a geometric sequence of real numbers, the sum of the first
\n" ); document.write( "two terms is 7 and the sum of the first six terms is 91. What is the sum of
\n" ); document.write( "the first four terms?
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document.write( "We are given\r\n" );
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document.write( "     +  = 7    (1)\r\n" );
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document.write( "     +  +  +  +  +  = 91      (2)   \r\n" );
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document.write( "In terms of \"a\" (the first term) and \"r\" (the common ratio) these equalities take the form\r\n" );
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document.write( "    a + ar = 7,       (3)\r\n" );
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document.write( "    a + ar +  +  +  +   = 91    (4)\r\n" );
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document.write( "In (4), group the terms\r\n" );
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document.write( "    (a + ar) + ( + ) + ( + )  = 91.    (5)\r\n" );
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document.write( "Re-write (5) in an equivalent form\r\n" );
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document.write( "    (a + ar) +  + ) = 91.    (6)\r\n" );
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document.write( "In (6), replace (a+ar) by the value of 7, based on (3).  You will get\r\n" );
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document.write( "    7 +  +  = 91.   (7)\r\n" );
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document.write( "In (7), divide both sides by 7\r\n" );
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document.write( "    1 +  +  = 13,\r\n" );
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document.write( "or\r\n" );
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document.write( "     +  - 12 = 0.    (8)\r\n" );
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document.write( "This biquadratic equation (8) is the quadratic equation relative .  Solve it by factoring\r\n" );
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document.write( "     = 0\r\n" );
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document.write( "Since we are given that the progression is in real numbers,   can not be zero \r\n" );
document.write( "(it is positive at any real r), we conclude that only possible value of    is\r\n" );
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document.write( "     = 3.\r\n" );
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document.write( "Then for the sum of the first four terms of this geometric progression    we have\r\n" );
document.write( "     = a + ar +  +  = (a + ar) + ( + ) = (a+ar) + .\r\n" );
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document.write( "We substitute here  a+ar = 7 and  = 3, and we get\r\n" );
document.write( "     = 7 + 3*7 = 7 + 21 = 28.\r\n" );
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document.write( "ANSWER.  The sum of the first 4 terms of this GP is 28.\r\n" );
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