document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=172794>Question 172794</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the sum  \r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #127636 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=127636\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Note: there are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=305-150%2B1=156 ALIGN=MIDDLE  ALT=\"305-150%2B1=156\"   DISABLED_style= \"max-width:90%; height:auto;\" > terms that need to be added.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sum%284i-5%2Ci=150%2C305%29 ALIGN=MIDDLE  ALT=\"sum%284i-5%2Ci=150%2C305%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given summation\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sum%284i%2Ci=150%2C305%29%2Bsum%28-5%2Ci=150%2C305%29 ALIGN=MIDDLE  ALT=\"sum%284i%2Ci=150%2C305%29%2Bsum%28-5%2Ci=150%2C305%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Break up the summation\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sum%284i%2Ci=150%2C305%29%2B%28-5%29%2A%28156%29 ALIGN=MIDDLE  ALT=\"sum%284i%2Ci=150%2C305%29%2B%28-5%29%2A%28156%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Since the 2nd term has no variables, we can simply multiply the inner value -5 by the number of terms 156\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sum%284i%2Ci=150%2C305%29-780 ALIGN=MIDDLE  ALT=\"sum%284i%2Ci=150%2C305%29-780\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2Asum%28i%2Ci=150%2C305%29-780 ALIGN=MIDDLE  ALT=\"4%2Asum%28i%2Ci=150%2C305%29-780\"   DISABLED_style= \"max-width:90%; height:auto;\" > Pull out the coefficient 4\r<BR>\n" );
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document.write( "Now this is where the formula <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sum%28i%2Ci=1%2Cn%29=%28n%28n%2B1%29%29%2F2 ALIGN=MIDDLE  ALT=\"sum%28i%2Ci=1%2Cn%29=%28n%28n%2B1%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > comes in. There's one small problem though: the terms NEED to start at 1. So what this means is that we have to break <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sum%28i%2Ci=150%2C305%29 ALIGN=MIDDLE  ALT=\"sum%28i%2Ci=150%2C305%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > into \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sum%28i%2Ci=150%2C305%29=sum%28i%2Ci=1%2C305%29-sum%28i%2Ci=1%2C149%29 ALIGN=MIDDLE  ALT=\"sum%28i%2Ci=150%2C305%29=sum%28i%2Ci=1%2C305%29-sum%28i%2Ci=1%2C149%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sum%28i%2Ci=150%2C305%29=%28305%28305%2B1%29%29%2F2-%28149%28149%2B1%29%29%2F2 ALIGN=MIDDLE  ALT=\"sum%28i%2Ci=150%2C305%29=%28305%28305%2B1%29%29%2F2-%28149%28149%2B1%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Now use the formula <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sum%28i%2Ci=1%2Cn%29=%28n%28n%2B1%29%29%2F2 ALIGN=MIDDLE  ALT=\"sum%28i%2Ci=1%2Cn%29=%28n%28n%2B1%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sum%28i%2Ci=150%2C305%29=%28305%28306%29%29%2F2-%28149%28150%29%29%2F2 ALIGN=MIDDLE  ALT=\"sum%28i%2Ci=150%2C305%29=%28305%28306%29%29%2F2-%28149%28150%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Add\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sum%28i%2Ci=150%2C305%29=305%28153%29-149%2875%29 ALIGN=MIDDLE  ALT=\"sum%28i%2Ci=150%2C305%29=305%28153%29-149%2875%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Reduce\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sum%28i%2Ci=150%2C305%29=46665-11175 ALIGN=MIDDLE  ALT=\"sum%28i%2Ci=150%2C305%29=46665-11175\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sum%28i%2Ci=150%2C305%29=35490 ALIGN=MIDDLE  ALT=\"sum%28i%2Ci=150%2C305%29=35490\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract\r<BR>\n" );
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document.write( "So the expression \r<BR>\n" );
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document.write( "then becomes\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2A%2835490%29-780 ALIGN=MIDDLE  ALT=\"4%2A%2835490%29-780\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=141960-780 ALIGN=MIDDLE  ALT=\"141960-780\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=141180 ALIGN=MIDDLE  ALT=\"141180\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract\r<BR>\n" );
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document.write( "So this means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sum%284i-5%2Ci=150%2C305%29=141180 ALIGN=MIDDLE  ALT=\"sum%284i-5%2Ci=150%2C305%29=141180\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
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