document.write( "Question 840912: The first two terms of a geometric sequence and an arithmetic sequence are the same. The first term is 12. The sum of the first three terms of the geometric sequence is 3 more than the sum of the first three terms of the arithmetic sequence. Determine TWO possible values for the common ratio, r, of the geometric sequence \n" ); document.write( "
Algebra.Com's Answer #506491 by Edwin McCravy(20056)\"\" \"About 
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The first two terms of a geometric sequence and an arithmetic sequence are the same. The first term is 12. The sum of the first three terms of the geometric sequence is 3 more than the sum of the first three terms of the arithmetic sequence. Determine TWO possible values for the common ratio, r, of the geometric sequence
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document.write( "Geometric sequence = a,ar,arē,__,__,...\r\n" );
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document.write( "Arithmetic sequence= a,a+d,a+2d,__,__,...
The first two terms of a geometric sequence and an arithmetic sequence are the same.
a = a, ar = a+d\r\n" );
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document.write( "The first term is 12. \r\n" );
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document.write( "a = 12,  12r = 12+d
The sum of the first three terms of the geometric sequence is 3 more than the sum of the first three terms of the arithmetic sequence.
   (a)+(ar)+(arē) = (a)+(a+d)+(a+2d)+3\r\n" );
document.write( "(12)+(12r)+(12rē) = (12)+(12+d)+(12+2d)+3\r\n" );
document.write( "      12+12r+12rē = 12+12+d+12+2d+3\r\n" );
document.write( "      12+12r+12rē = 39+3d\r\n" );
document.write( "         12r+12rē = 27+3d\r\n" );
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document.write( "Every term can be divided by 3\r\n" );
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document.write( "           4r+4rē = 9+d\r\n" );
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document.write( "So we have this system of two equations and 2 unknowns:\r\n" );
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document.write( "     12r = 12+d, 4r+4rē = 9+d\r\n" );
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document.write( "Solve the first for d:\r\n" );
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document.write( "         12r = 12+d     \r\n" );
document.write( "      12r-12 = d\r\n" );
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document.write( "Substitute in\r\n" );
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document.write( "      4r+4rē = 9+d\r\n" );
document.write( "      4r+4rē = 9+(12r-12)\r\n" );
document.write( "      4r+4rē = 9+12r-12\r\n" );
document.write( "      4r+4rē = 12r-3\r\n" );
document.write( "    4rē-8r+3 = 0\r\n" );
document.write( "(2r-1)(2r-3) = 0\r\n" );
document.write( "   2r-1=0;  2r-3=0\r\n" );
document.write( "     2r=1;    2r=3\r\n" );
document.write( "      r=\"1%2F2\";    r=\"3%2F2\"\r\n" );
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document.write( "Those are the two values.\r\n" );
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document.write( "To check we must find d\r\n" );
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document.write( "Using r = \"1%2F2\"\r\n" );
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document.write( " 12r-12 = d\r\n" );
document.write( "12\"%281%2F2%29\"-12 = d\r\n" );
document.write( "   6-12 = d\r\n" );
document.write( "     -6 = d\r\n" );
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document.write( "Arithmetic sequence:  12, 6, 0, ...\r\n" );
document.write( "Geometric sequence:   12, 6, 3, ...\r\n" );
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document.write( "Sum of 1st three terms of arithmetic sequence = 12+6+0 = 18\r\n" );
document.write( "Sum of 1st three terms of geometric sequence  = 12+6+3 = 21\r\n" );
document.write( "21 is 3 more than 18\r\n" );
document.write( "That checks.\r\n" );
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document.write( "Using r = \"3%2F2\"\r\n" );
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document.write( " 12r-12 = d\r\n" );
document.write( "12\"%283%2F2%29\"-12 = d\r\n" );
document.write( "  18-12 = d\r\n" );
document.write( "      6 = d\r\n" );
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document.write( "Arithmetic sequence:  12, 18, 24, ...\r\n" );
document.write( "Geometric sequence:   12, 18, 27, ...\r\n" );
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document.write( "Sum of 1st three terms of arithmetic sequence = 12+18+24 = 54\r\n" );
document.write( "Sum of 1st three terms of geometric sequence  = 12+18+27 = 57\r\n" );
document.write( "57 is 3 more than 54\r\n" );
document.write( "That checks, too.  So it's correct\r\n" );
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document.write( "Edwin
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