document.write( "Question 1103432: Find the common ratio of a finite geometric series if the first term is 11, and the sum of the first 12 terms is 2922920. \n" ); document.write( "

Algebra.Com's Answer #718164 by ikleyn(52794) $\"\"$ $\"About$

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document.write( "Let  \"r\" be the unknown common term.\r\n" );
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document.write( "Then\r\n" );
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document.write( "S =  = 2922920,  which implies\r\n" );
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document.write( "1 + r + r^2 + r^3 + . . . + r^11 =  = 265720,   or\r\n" );
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document.write( " = 265720\r\n" );
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document.write( "You can check that r= 3 is the solution to this equation:   = 265720.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Answer. The common ratio of this progression is 3.\r
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document.write( "It is not difficult to prove that r= 3 is the UNIQUE solution to the problem.\r\n" );
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document.write( "Indeed, if r >=1 then the sum  1 + r + r^2 + . . . + r^11 is monotonic function of r.\r\n" );
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document.write( "Next, if  0 < r < 1, then this sum is less than 12.\r\n" );
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document.write( "Further, if  -1 <= r < 0, then AGAIN this sum is less than 12.\r\n" );
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document.write( "Finally, if  r < -1,  then    has POSITIVE numerator and negative denominator, which means that this rational function is NEGATIVE.\r\n" );
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document.write( "The plot below ILLUSTRATES this behavior of the function  1 + r + r^2 + . . . + r^11.\r\n" );
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document.write( "Plot y =  (red)  and y = 265720 (green)\r\n" );
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