document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=929332>Question 929332</A>: <I><SPAN STYLE=\"word-break: break-all;\"> please help me find the sum of the following sequence (-6)+(-1)+4+9+...+39 thank you </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #564206 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=564206\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The sum (-6)+(-1)+4+9+...+39 is the sum of an arithmetic sequence<BR>\n" );
document.write( "whose first term is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-6%29 ALIGN=MIDDLE  ALT=\"%28-6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and each term is equal to the term before plus the common difference: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5 ALIGN=MIDDLE  ALT=\"5\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "For a timed test, I would solve the problem by mental math to save time.<BR>\n" );
document.write( "If a pesky teacher insists on \"show your work\", I would go the formula-happy way.\r<BR>\n" );
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document.write( "THE MENTAL MATH WAY:<BR>\n" );
document.write( "How many terms?<BR>\n" );
document.write( "There are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10 ALIGN=MIDDLE  ALT=\"10\"   DISABLED_style= \"max-width:90%; height:auto;\" > terms.<BR>\n" );
document.write( "There are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" > negative terms,<BR>\n" );
document.write( "and the positive terms go from <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4=5%2A1=1%2A5-1 ALIGN=MIDDLE  ALT=\"4=5%2A1=1%2A5-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=39=40-1=8%2A5-1 ALIGN=MIDDLE  ALT=\"39=40-1=8%2A5-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "so without listing and counting, I know there are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8 ALIGN=MIDDLE  ALT=\"8\"   DISABLED_style= \"max-width:90%; height:auto;\" > positive terms.<BR>\n" );
document.write( "So the total number of terms is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8%2B2=10 ALIGN=MIDDLE  ALT=\"8%2B2=10\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The sum is the average of the terms times the number of terms,<BR>\n" );
document.write( "but in an arithmetic sequence,<BR>\n" );
document.write( "the average of the terms is the same as the average of the first and last terms<BR>\n" );
document.write( "(which is the sum of the first and last terms divided by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" > ).<BR>\n" );
document.write( "That sum is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-6%29%2B39=33 ALIGN=MIDDLE  ALT=\"%28-6%29%2B39=33\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and that sum times <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10 ALIGN=MIDDLE  ALT=\"10\"   DISABLED_style= \"max-width:90%; height:auto;\" > and divided by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=33%2A10%2F2=highlight%28165%29 ALIGN=MIDDLE  ALT=\"33%2A10%2F2=highlight%28165%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "THE FORMULA-HAPPY WAY:<BR>\n" );
document.write( "First we make sure that it is an arithmetic sequence, meaning that there is a common difference, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d ALIGN=MIDDLE  ALT=\"d\"   DISABLED_style= \"max-width:90%; height:auto;\" > , between consecutive terms:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-1%29-%28-6%29=5 ALIGN=MIDDLE  ALT=\"%28-1%29-%28-6%29=5\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4-%28-1%29=5 ALIGN=MIDDLE  ALT=\"4-%28-1%29=5\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9-4=5 ALIGN=MIDDLE  ALT=\"9-4=5\"   DISABLED_style= \"max-width:90%; height:auto;\" > , so it is an arithmetic sequence,<BR>\n" );
document.write( "and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d=5 ALIGN=MIDDLE  ALT=\"d=5\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The first term is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B0%5D=-6 ALIGN=MIDDLE  ALT=\"a%5B0%5D=-6\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "We know that, in an arithmetic sequence, term number <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n ALIGN=MIDDLE  ALT=\"n\"   DISABLED_style= \"max-width:90%; height:auto;\" > is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5Bn%5D=a%5B0%5D%2B%28n-1%29%2Ad ALIGN=MIDDLE  ALT=\"a%5Bn%5D=a%5B0%5D%2B%28n-1%29%2Ad\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "so we apply that formula to term <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=39 ALIGN=MIDDLE  ALT=\"39\"   DISABLED_style= \"max-width:90%; height:auto;\" > , to find what is its term number, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n ALIGN=MIDDLE  ALT=\"n\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=39=-6%2B%28n-1%29%2A5 ALIGN=MIDDLE  ALT=\"39=-6%2B%28n-1%29%2A5\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=39%2B6=%28n-1%29%2A5 ALIGN=MIDDLE  ALT=\"39%2B6=%28n-1%29%2A5\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=45=%28n-1%29%2A5 ALIGN=MIDDLE  ALT=\"45=%28n-1%29%2A5\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=45%2F5=n-1 ALIGN=MIDDLE  ALT=\"45%2F5=n-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9=n-1 ALIGN=MIDDLE  ALT=\"9=n-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9%2B1=n ALIGN=MIDDLE  ALT=\"9%2B1=n\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n=10 ALIGN=MIDDLE  ALT=\"n=10\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Now we apply a formula to find<BR>\n" );
document.write( "the sum <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s%5Bn%5D ALIGN=MIDDLE  ALT=\"s%5Bn%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > of the first <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n=10 ALIGN=MIDDLE  ALT=\"n=10\"   DISABLED_style= \"max-width:90%; height:auto;\" > terms of an arithmetic sequence with a[0]=-6}}} and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d=5 ALIGN=MIDDLE  ALT=\"d=5\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "With sum luck, we were given the formula<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S%5Bn%5D=n%2A%28a%5B0%5D%2Ba%5Bn%5D%29%2F2 ALIGN=MIDDLE  ALT=\"S%5Bn%5D=n%2A%28a%5B0%5D%2Ba%5Bn%5D%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "If so, we apply it and get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S%5B10%5D=10%2A%28%28-6%29%2B39%29%2F2=10%2A33%2F2=330%2F2=highlight%28165%29 ALIGN=MIDDLE  ALT=\"S%5B10%5D=10%2A%28%28-6%29%2B39%29%2F2=10%2A33%2F2=330%2F2=highlight%28165%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Without luck, we may only have another more cumbersome formula, less convenient for this particular problem, like<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S%5Bn%5D=n%2A%282%2Aa%5B0%5D%2B%28n-1%29%2Ad%29%2F2 ALIGN=MIDDLE  ALT=\"S%5Bn%5D=n%2A%282%2Aa%5B0%5D%2B%28n-1%29%2Ad%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and we could apply that formula to get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S%5B10%5D=10%2A%282%2A%28-6%29%2B%2810-1%29%2A5%29%2F2=10%2A%28-12%2B9%2A5%29%2F2=10%2A%28-12%2B45%29%2F2=10%2A33%2F2=330%2F2=highlight%28165%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > . </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );