document.write( "Question 955519: If the sum of first four terms of an arithmetic sequence is 16 and their product 105 find the terms if they all are integers ... I know that the A.s is (1,3,5,7) but how to get it \n" ); document.write( "
Algebra.Com's Answer #584450 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "If the first term is \"a\" and the difference is \"d\", then the first four terms \r\n" );
document.write( "are \r\n" );
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document.write( "a, a+d, a+2d, a+3d \r\n" );
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\n" ); document.write( "the sum of first four terms of an arithmetic sequence is 16
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document.write( "So, \r\n" );
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document.write( "a + a+d + a+2d + a+3d = 16\r\n" );
document.write( "              4a + 6d = 16 \r\n" );
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document.write( "Divide thru by 2 \r\n" );
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document.write( "              2a + 3d = 8 \r\n" );
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\n" ); document.write( "their product 105,
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document.write( "So  \r\n" );
document.write( "   a(a+d)(a+2d)(a+3d) = 105 \r\n" );
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document.write( "So we have this system of two equations: \r\n" );
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document.write( "\"system%282a+%2B+3d+=+8%2Ca%28a%2Bd%29%28a%2B2d%29%28a%2B3d%29+=+105%29\" \r\n" );
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document.write( "Solve the first equation for \"a\":\r\n" );
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document.write( "2a = 8-3d\r\n" );
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document.write( " a = \"%288-3d%29%2F2\"\r\n" );
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document.write( "Substitute in the second equation:\r\n" );
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document.write( "\"a%28a%2Bd%29%28a%2B2d%29%28a%2B3d%29+=+105\"\r\n" );
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document.write( "Get LCD in each parentheses:\r\n" );
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document.write( "Combine fractions in parentheses:\r\n" );
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document.write( "\"%28%288-3d%29%2F2%29%28%288-d%29%2F2%29%28%288%2Bd%29%2F2%29%28%288%2B3d%29%2F2%29+=+105\"\r\n" );
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document.write( "Multiply both sides by 16\r\n" );
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document.write( "\"%288-3d%29%288-d%29%288%2Bd%29%288%2B3d%29+=+1680\"\r\n" );
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document.write( "Rearrange the factors on the left:\r\n" );
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document.write( "\"%288-3d%29%288%2B3d%29%288-d%29%288%2Bd%29+=+1680\"\r\n" );
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document.write( "FOIL\r\n" );
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document.write( "\"%2864-9d%5E2%29%2864-d%5E2%29+=+1680\"\r\n" );
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document.write( "FOIL again:\r\n" );
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document.write( "\"4096-640d%5E2%2B9d%5E4=1680\"\r\n" );
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document.write( "Get 0 on the right side:\r\n" );
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document.write( "\"2416-640d%5E2-9d%5E4=0\"\r\n" );
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document.write( "Get in descending order:\r\n" );
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document.write( "\"9d%5E4-640d%5E2%2B2416=0\"\r\n" );
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document.write( "Factor the trinomial:\r\n" );
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document.write( "\"%289d%5E2-604%29%28d%5E2-4%29=0\"\r\n" );
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document.write( "Factor further:\r\n" );
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document.write( "\"%289d%5E2-604%29%28d-2%29%28d%2B2%29=0\"\r\n" );
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document.write( "\"9d%5E2-604=0\", \"d-2=0\", \"d%2B2=0\"\r\n" );
document.write( "\"9d%5E2=604\", \"d=2\", \"d=-2\"\r\n" );
document.write( "\"d%5E2=604%2F9\"\r\n" );
document.write( "\"d=+%22%22+%2B-+sqrt%28604%2F9%29\"\r\n" );
document.write( "\"d=+%22%22+%2B-+sqrt%284%2A151%29%2F3\"\r\n" );
document.write( "\"d=+%22%22+%2B-+2sqrt%28151%29%2F3\"\r\n" );
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document.write( "We can ignore the irrational values for d, since the sequence \r\n" );
document.write( "has only integers.\r\n" );
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document.write( "The only integer values for d are d=2 and d=-2\r\n" );
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document.write( "For d=2, substitute in\r\n" );
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document.write( " a = \"%288-3d%29%2F2\"\r\n" );
document.write( " a = \"%288-3%2A2%29%2F2\"\r\n" );
document.write( " a = \"%288-6%29%2F2\"\r\n" );
document.write( " a = \"2%2F2\"\r\n" );
document.write( " a = \"1\"\r\n" );
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document.write( "Substitute d=2 and a=1 in\r\n" );
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document.write( "a, a+d, a+2d, a+3d\r\n" );
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document.write( "1, 1+2, 1+2(2), 1+3(2)\r\n" );
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document.write( "Therefore 1 solution is\r\n" );
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document.write( "1, 3, 5, 7\r\n" );
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document.write( "----------------------------\r\n" );
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document.write( "For d=-2, substitute in\r\n" );
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document.write( " a = \"%288-3d%29%2F2\"\r\n" );
document.write( " a = \"%288-3%2A%28-2%29%29%2F2\"\r\n" );
document.write( " a = \"%288%2B6%29%2F2\"\r\n" );
document.write( " a = \"14%2F2\"\r\n" );
document.write( " a = \"7\"\r\n" );
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document.write( "Substitute d=-2 and a=7 in\r\n" );
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document.write( "a, a+d, a+2d, a+3d\r\n" );
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document.write( "7, 7+(-2), 7+2(-2), 7+3(-2)\r\n" );
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document.write( "7, 5, 7-4, 7-6\r\n" );
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document.write( "Therefore another solution is\r\n" );
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document.write( "7, 5, 3, 1.\r\n" );
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document.write( "But this is the same sequence reversed.\r\n" );
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document.write( "Edwin
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