Algebra.Com's Answer #754606 by ikleyn(52781)![\"\"](\"/images/stars/stars-13.gif\")  
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document.write( " AM is the standard abbreviation for \"Arithmetic Mean\".\r
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document.write( " GM is the standard abbreviation for \"Geometric Mean\".\r
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document.write( " So, almost everything is clear, except one thing: When the condition says \"the ratio of the AM to the GM is 1:-2\"\r
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document.write( " it missed to say to which progressions these AM and GM do relate.\r
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document.write( " So, it requires some interpretation; therefore I will re-edit the problem in this way:\r
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document.write( " The sums of the first three terms of an AP and a GP are equal and non-zero,\r\n" );
document.write( " the common difference of the AP and the common ratio of the GP are equal, \r\n" );
document.write( " and the ratio of the AM of AP to the GM of GP is 1:-2.\r\n" );
document.write( " Find the common ratio of the GP.\r\n" );
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document.write( " by underlying my insertions.\r
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document.write( "Solution\r
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document.write( "Since this problem, although elementary, is a bit higher than the average School Math level, I will assume that the reader's
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document.write( "level does correspond to the problem level, and will not explain elementary details.\r
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document.write( "Let \" a \" be the middle (the second) term of the AP and \" b \" is the middle (the second) term of the GP.\r\n" );
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document.write( "Then it is widely known that AM of the first three terms of an AP is equal to \"a\", \r\n" );
document.write( "while the GM of the first three terms of an GP is equal to \"b\".\r\n" );
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document.write( "Therefore this statement of the condition \"the ratio of the AM of AP to the GM of GP is 1:-2\" simply means that \r\n" );
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document.write( "  =  =  . (1)\r\n" );
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document.write( "Let \"r\" be the common difference of the AP and the common ratio of the GP (they are equal, so I use one symbol for both).\r\n" );
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document.write( "Then the fact that \"The sums of the first three terms of an AP and a GP are equal\"\r\n" );
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document.write( "gives the second equation\r\n" );
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document.write( " 3a =  + b +  . (2)\r\n" );
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document.write( "From equation (1), express b = -2a and substitute it into equation (2). You will get\r\n" );
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document.write( " 3a =  - 2a - (2a)*r.\r\n" );
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document.write( "Cancel factor \"a\" in both sides\r\n" );
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document.write( " 3 =  - 2 - 2r;\r\n" );
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document.write( "Multiply by \"r\" both sides\r\n" );
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document.write( " 3r = - 2 - 2r - 2r^2\r\n" );
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document.write( "and write the quadratic equation in the standard form\r\n" );
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document.write( " 2r^2 + 5r + 2 = 0.\r\n" );
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document.write( "Quadratic formula gives the roots r = -2 and r = .\r\n" );
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document.write( "So, the common difference of the AP may have these two values,\r\n" );
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document.write( "and the common difference of the GP may have these two values, RESPECTIVELY.\r\n" );
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document.write( "Having these solutions for \"r\", we can make one step forward and find \"the relationship between first terms of the AP and GP\".\r\n" );
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document.write( "a) If a = r = -2, then  = a - r = a - 2;  =  = substitute here b = -2a from above =  = a.\r\n" );
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document.write( " Therefore, = a = + 2 is that relationship between and in this case.\r\n" );
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document.write( "b) If a = r = , then = a - r = a +  ; =  = substitute here b = -2a from above =  = 4a.\r\n" );
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document.write( " Therefore, = 4a =  - 2 is that relationship between and in this case.\r\n" );
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document.write( "ANSWER. There are two possibilities: a) r = -2 and = + 2; and\r\n" );
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document.write( " b) r = and = -2.\r\n" );
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document.write( "Solved.\r
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