document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1099657>Question 1099657</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The odd integers are arranged into sets, each succeeding set continuing from the previous, and containing one more number as follows: {1},{3,5},{7,9,11},{13,15,17,19},... what is the sum of the numbers in the set that has last number 1259? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #714121 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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/cgi-bin/show-face.mpl?user=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=714121\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "The numbers of numbers in the successive sets are<BR>\n" );
document.write( "1, 2, 3, 4, ...<br><BR>\n" );
document.write( "Let's think about the process for finding the answer by answering a similar but much simpler problem: find the sum of the numbers in the set that has last number 19.<br><BR>\n" );
document.write( "We can see the answer in the statement of the problem; however, let's see how we can get the answer in a way that we can use to find the answer to the given problem.<br><BR>\n" );
document.write( "The \"first\" set contains 1 odd integer; the \"second\" contains 2; the third 3; and the fourth 4.<br><BR>\n" );
document.write( "1+2+3+4 = 10, and 19 is the 10th odd integer, so 19 is the last number in the 4th set.<br><BR>\n" );
document.write( "Then, since there are 4 numbers in that set, the four numbers in the set are 19, 17, 15, and 13.<br><BR>\n" );
document.write( "It is easy to see in this case that the sum of those numbers is 64.  But let's find that sum by a method that we can use for the given problem, where we can't see the whole set of numbers at once.<br><BR>\n" );
document.write( "We know at this point that 19 is the largest number in the 4th set.  The smallest number in the 4th set is 19 minus the common difference 2 three times, so the smallest number in the set is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=19-3%282%29+=+19-6+=+13 ALIGN=MIDDLE  ALT=\"19-3%282%29+=+19-6+=+13\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  Then the average of the numbers in the set is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2819%2B13%29%2F2+=+16 ALIGN=MIDDLE  ALT=\"%2819%2B13%29%2F2+=+16\"   DISABLED_style= \"max-width:90%; height:auto;\" >; and with 4 numbers in the set, the sum is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16%2A4+=+64 ALIGN=MIDDLE  ALT=\"16%2A4+=+64\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<br><BR>\n" );
document.write( "Now let's apply that same analysis to the given problem.<br><BR>\n" );
document.write( "The number 1259 is the 630th odd positive integer; and we are told that it is the last number in its set.<br><BR>\n" );
document.write( "So we need to find when 1+2+3+4+...+n is equal to 630.<br><BR>\n" );
document.write( "The sum of the integers from 1 to n is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28n%28n%2B1%29%29%2F2 ALIGN=MIDDLE  ALT=\"%28n%28n%2B1%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >; so we want to find a solution to<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28n%28n%2B1%29%29%2F2+=+630 ALIGN=MIDDLE  ALT=\"%28n%28n%2B1%29%29%2F2+=+630\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%28n%2B1%29+=+1260 ALIGN=MIDDLE  ALT=\"n%28n%2B1%29+=+1260\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "By one method or another, we find n=35.<br><BR>\n" );
document.write( "That means 1259 is the last number in the 35th set.<br><BR>\n" );
document.write( "Using the method we developed earlier, the first number in the 35th set is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1259-34%2A2+=+1259-68+=+1191 ALIGN=MIDDLE  ALT=\"1259-34%2A2+=+1259-68+=+1191\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<br><BR>\n" );
document.write( "Then the average of the numbers in the set is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281259%2B1191%29%2F2+=+2450%2F2+=+1225 ALIGN=MIDDLE  ALT=\"%281259%2B1191%29%2F2+=+2450%2F2+=+1225\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<br><BR>\n" );
document.write( "And then the sum of the numbers in that set is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=35%2A1225+=+42875 ALIGN=MIDDLE  ALT=\"35%2A1225+=+42875\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<br><BR>\n" );
document.write( "ANSWER:  The sum of the numbers in the set with last number 1259 is 42,875.<BR>\n" );
document.write( " </SPAN>\n" );
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