document.write( "Question 1190567: Please help: the sum of the first two terms of a G.P is x.The sum of the last two is 1 if there are n term in all, calculate the common ratio \n" ); document.write( "

Algebra.Com's Answer #822268 by ikleyn(52793) $\"\"$ $\"About$

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\n" ); document.write( "Please help: the sum of the first two terms of a G.P is x.The sum of the last two is 1.
\n" ); document.write( "if there are n term in all, calculate the common ratio
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document.write( "Let the GP be a, ar, ar^2, . . . , ar^(n-1).\r\n" );
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document.write( "The first term is \"a\", the common ratio is \"r\"; the number of terms is \"n\".\r\n" );
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document.write( "We are given \r\n" );
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document.write( "    (a)  the value of n;            (the number of terms);\r\n" );
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document.write( "    (b)  a + ar = x;                (the sum of the 1st and the 2nd terms, x)\r\n" );
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document.write( "    (c)  ar^(n-2) + ar^(n-1) = 1.   (the sum of the two last terms, 1).\r\n" );
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document.write( "From (a), we have\r\n" );
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document.write( "    a*(1+r) = x.               (1)\r\n" );
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document.write( "From (b), we have\r\n" );
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document.write( "     = 1.      (2)\r\n" );
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document.write( "Now divide equation (2) by equation (1). You will get, after canceling common factors\r\n" );
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document.write( "     = .\r\n" );
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document.write( "It implies  \r\n" );
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document.write( "    r =  =   = .       ANSWER\r\n" );
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document.write( "Any of the equivalent forms in the line above is the answer.\r\n" );
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document.write( "So, the value of the common ratio \"r\" is      by the input data x and n.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Solved.\r
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As a conclusion

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\n" ); document.write( "\n" ); document.write( "The problem is posed correctly. It has (it admits) a unique solution.\r
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\n" ); document.write( "\n" ); document.write( "Tutor @math_tutor2020 treated the problem incorrectly. He missed part of input data.\r
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\n" ); document.write( "\n" ); document.write( "Therefore, his solution is incorrect.\r
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\n" ); document.write( "\n" ); document.write( "His diagnosis \r
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\n" ); document.write( "\n" ); document.write( " \"there are infinitely many possible solutions here.
\n" ); document.write( " There isn't enough info to answer this current question.\"\r
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\n" ); document.write( "\n" ); document.write( "is incorrect.\r
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\n" ); document.write( "\n" ); document.write( "The problem is posed correctly, and I gave a complete solution/analysis.\r
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