document.write( "Question 1099272: How many different integers can be represented as a sum of four distinct numbers chosen from the set ? \n" ); document.write( "
Algebra.Com's Answer #713718 by Edwin McCravy(20056)\"\" \"About 
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document.write( "The elements of the set {6,17,28,39,...,105} are terms of the \r\n" );
document.write( "arithmetic sequence with first term 6, last term 105, and common\r\n" );
document.write( "difference 11.\r\n" );
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document.write( "So the nth term is \"a%5Bn%5D=a%5B1%5D%2B%28n-1%29d=6%2B%28n-1%29%2811%29=6%2B11n-11=11n-5\"\r\n" );
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document.write( "The number of terms is found by setting 11n-5=105 and solving for n\r\n" );
document.write( "                                        11n=110\r\n" );
document.write( "                                          n=10 \r\n" );
document.write( "So there are 10 terms.\r\n" );
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document.write( "Suppose the pth, qth, rth, and sth terms are the 4 distinct terms. \r\n" );
document.write( "Then their sum is given by: \r\n" );
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document.write( "ap + aq + ar + as =\r\n" );
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document.write( "(11p-5) + (11q-5) + (11r-5) + (11s-5) = 11p + 11q + 11r + 11s - 20 =\r\n" );
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document.write( "11(p + q + r + s)-20 = 11z-20 where z = p + q + r + s \r\n" );
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document.write( "the sum z = p + q + r + s can take on any integer value from the \r\n" );
document.write( "smallest sum 1 + 2 + 3 + 4 = 10 to the largest sum 7 + 8 + 9 + 10 = 34\r\n" );
document.write( "inclusive.  There are 34 integers from 1 to 34 inclusive.  There\r\n" );
document.write( "are 9 integers from 1 to 9, inclusive that we do not want to count. \r\n" );
document.write( "So there are 34-9 or 25 integers from 10 to 34 inclusive.\r\n" );
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document.write( "Each integer from 10 to 34 inclusive gives a different sum when \r\n" );
document.write( "substituted in 11z-20, so the answer is that 25 integers can be \r\n" );
document.write( "represented as a sum of 4 distinct integers chosen from \r\n" );
document.write( "{6,17,28,39,...,105}.\r\n" );
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document.write( "Edwin
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