document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1001423>Question 1001423</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please help me solve the problem\r<BR>\n" );
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document.write( "Let Sn(n>=1) be a sequence of sets defined by \r<BR>\n" );
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document.write( "S1 = {0}<BR>\n" );
document.write( "S2 ={3/2,5/2}<BR>\n" );
document.write( "S3={8/3,11/3,14/3}<BR>\n" );
document.write( "S4={15/4,19/4,23/4,27/4},.......... then\r<BR>\n" );
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document.write( "a) third element in S20 is 439/20<BR>\n" );
document.write( "b)third element in S20 is 431/20<BR>\n" );
document.write( "c)sum of element in S20 is 589<BR>\n" );
document.write( "d) sum of the elements in S20 is 609\r<BR>\n" );
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document.write( "[Sn denotes S subscript n eg. S1 denotes S subscript 1]\r<BR>\n" );
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document.write( "Which of the above options(a,b,c,d) are correct? {at least two of them must be correct} </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #618608 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=618608\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> S1 has 1 element<BR>\n" );
document.write( "S2 has 2 elements<BR>\n" );
document.write( "S3 has 3 elements<BR>\n" );
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document.write( "S20 has 20 elements<BR>\n" );
document.write( "Sn has n elements\r<BR>\n" );
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document.write( "The denominator of each term in S2 is 2<BR>\n" );
document.write( "The denominator of each term in S3 is 3<BR>\n" );
document.write( "The denominator of each term in S4 is 4<BR>\n" );
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document.write( "The denominator of each term in S20 is 20<BR>\n" );
document.write( "The denominator of each term in Sn is n\r<BR>\n" );
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document.write( "The numerators are a bit trickier to figure out. But notice how \r<BR>\n" );
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document.write( "S2 has numerators 3,5<BR>\n" );
document.write( "so we have +2 each time\r<BR>\n" );
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document.write( "S3 has numerators 8,11,14<BR>\n" );
document.write( "we're adding 3 each time\r<BR>\n" );
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document.write( "S4 has numerators 15,19,23,27<BR>\n" );
document.write( "we're adding 4 each time\r<BR>\n" );
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document.write( "so Sn will have us adding n each time<BR>\n" );
document.write( "The question is: where do we start?\r<BR>\n" );
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document.write( "S2 starts with 3<BR>\n" );
document.write( "S3 starts with 8<BR>\n" );
document.write( "S4 starts with 15\r<BR>\n" );
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document.write( "It turns out that the sequence 3,8,15 follows a quadratic pattern. In other words, a quadratic goes through the three points (2,3), (3,8) and (4,15)\r<BR>\n" );
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document.write( "I'm not going to go into too much detail about this (since it takes a very long time to do by hand), but you use a calculator to get this quadratic y = x^2 - 1\r<BR>\n" );
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document.write( "So Sn has n elements. The first element would be(n^2-1)/n\r<BR>\n" );
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document.write( "The first element of S20 would be...\r<BR>\n" );
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document.write( "(n^2-1)/n = (20^2-1)/20 = 399/20\r<BR>\n" );
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document.write( "Now we add 20 two times to get\r<BR>\n" );
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document.write( "399+20+20 = 439\r<BR>\n" );
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document.write( "So the third element of S20 is 439/20\r<BR>\n" );
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document.write( "We start with 399 in the numerator and we have the sequence\r<BR>\n" );
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document.write( "399, 419, 439, 479, ...\r<BR>\n" );
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document.write( "we have 20 of these terms being added up. This can be done quickly with an arithmetic series formula\r<BR>\n" );
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document.write( "sum of first n terms = n*(a1 + an)/2<BR>\n" );
document.write( "sum of first n terms = n*(a1 + a1 + d(n-1))/2<BR>\n" );
document.write( "sum of first n terms = n*(2*a1 + d(n-1))/2<BR>\n" );
document.write( "sum of first 20 terms = 20*(2*399 + 20(20-1))/2<BR>\n" );
document.write( "sum of first 20 terms = 11,780\r<BR>\n" );
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document.write( "The sum of the terms in the numerator is 11,780\r<BR>\n" );
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document.write( "The denominator stays as 20\r<BR>\n" );
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document.write( "11,780/20 = 589\r<BR>\n" );
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document.write( "Final Answers:\r<BR>\n" );
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document.write( "a) third element in S20 is 439/20<BR>\n" );
document.write( "c) sum of element in S20 is 589 </SPAN>\n" );
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