document.write( "Question 224627: The first three terms of an arithmetic series have a sum of 24 and a product of 312. What is the fourth term in the series? \n" ); document.write( "
Algebra.Com's Answer #167799 by Edwin McCravy(20056)\"\" \"About 
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The first three terms of an arithmetic series have a sum of 24 and a product of 512. What is the fourth term in the series?
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document.write( "Let: \r\n" );
document.write( "The common difference = d\r\n" );
document.write( "The first term = a\r\n" );
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document.write( "Therefore:\r\n" );
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document.write( "The second term = a+d\r\n" );
document.write( "The third term = a+d+d = a+2d\r\n" );
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document.write( ">>...The first three terms of an arithmetic series have a sum of 24...<<\r\n" );
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document.write( "           a + (a+d) + (a+2d) = 24\r\n" );
document.write( "                   a+a+d+a+2d = 24\r\n" );
document.write( "                        3a+3d = 24\r\n" );
document.write( "                          a+d = 8\r\n" );
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document.write( ">>...The first three terms of an arithmetic series have...a product of 512...<<\r\n" );
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document.write( "                 a(a+d)(a+2d) = 512  \r\n" );
document.write( "                 \r\n" );
document.write( "So we have to solve the system:\r\n" );
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document.write( "\"system%28a%2Bd=8%2Ca%28a%2Bd%29%28a%2B2d%29=512%29\"\r\n" );
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document.write( "Solve the first equation for d\r\n" );
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document.write( "\"d=8-a\"\r\n" );
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document.write( "Substitute in\r\n" );
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document.write( "\"a%28a%2Bd%29%28a%2B2d%29+=+312\"\r\n" );
document.write( "\"a%28a%2B%288-a%29%29%28a%2B2%288-a%29%29+=+312\"\r\n" );
document.write( "\"a%28a%2B8-a%29%28a%2B16-2a%29%29+=+312\"\r\n" );
document.write( "\"a%288%29%28-a%2B16%29+=+312\"\r\n" );
document.write( "\"8a%28-a%2B16%29+=+312\"\r\n" );
document.write( "Divide both sides by 8\r\n" );
document.write( "\"a%28-a%2B16%29=39%0D%0A%7B%7B%7B-a%5E2%2B16a=39\"\r\n" );
document.write( "\"-a%5E2%2B16a-39=0\"\r\n" );
document.write( "Multiply through by -1\r\n" );
document.write( "\"a%5E2-16a%2B39=0\"\r\n" );
document.write( "Factor:\r\n" );
document.write( "\"%28a-13%29%28a-3%29=0\"\r\n" );
document.write( "Set each factor = 0\r\n" );
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document.write( "\"a-13=0\",     \"a-3=0\"\r\n" );
document.write( "\"a=13\"      \"a=3\"\r\n" );
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document.write( "There are two solutions for the first term, a,\r\n" );
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document.write( "Now since \"d=8-a\", taking the first solution:\r\n" );
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document.write( "When \"a=13\", \"d=8-a=8-%2813%29=8-13=-5\"\r\n" );
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document.write( "so the first three terms are\r\n" );
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document.write( "\"13\", \"13%2B%28-5%29=8\", \"8%2B%28-5%29=3\"\r\n" );
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document.write( "Checking:  their sum = 13+8+3=24\r\n" );
document.write( "           their product = (13)(8)(3)=312\r\n" );
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document.write( "So the fourth term is the third term plus (-5), or\r\n" );
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document.write( "fourth term = 3-5 = -2\r\n" );
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document.write( "-----------------------------\r\n" );
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document.write( "As before, since \"d=8-a\", taking the second solution:\r\n" );
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document.write( "When \"a=3\", \"d=8-a=8-%2813%29=8-3=5\"\r\n" );
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document.write( "so the first three terms are\r\n" );
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document.write( "\"3\", \"3%2B5=8\", \"8%2B5=13\"\r\n" );
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document.write( "Checking:  their sum = 3+8+13=24\r\n" );
document.write( "           their product = (3)(8)(13)=312\r\n" );
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document.write( "So the fourth term is the third term plus 5, or\r\n" );
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document.write( "fourth term = 13+5 = 18\r\n" );
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document.write( "So there are two solutions for the fourth term: -2, and 18\r\n" );
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document.write( "Edwin
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