document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1070612>Question 1070612</A>: <I><SPAN STYLE=\"word-break: break-all;\"> the sum of n terms of an arithmetic series is:Sn=5n^2-11. Determine common difference </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #685696 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=685696\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> There is a typo there.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S%5Bn%5D=5n%5E2-11 ALIGN=MIDDLE  ALT=\"S%5Bn%5D=5n%5E2-11\"   DISABLED_style= \"max-width:90%; height:auto;\" > cannot be the sum of n terms of an arithmetic series,<BR>\n" );
document.write( "but <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S%5Bn%5D=5n%5E2-11n ALIGN=MIDDLE  ALT=\"S%5Bn%5D=5n%5E2-11n\"   DISABLED_style= \"max-width:90%; height:auto;\" > could be .<BR>\n" );
document.write( "You could use either expression to find the common difference <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d=10 ALIGN=MIDDLE  ALT=\"d=10\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "but only a sum of the form <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=an%5E2%2Bbn ALIGN=MIDDLE  ALT=\"an%5E2%2Bbn\"   DISABLED_style= \"max-width:90%; height:auto;\" > could work to find an actual<BR>\n" );
document.write( "arithmetic series, with a first term <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B1%5D ALIGN=MIDDLE  ALT=\"a%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > independent of <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( " <BR>\n" );
document.write( "A WAY TO SOLVE IT WITH <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S%5Bn%5D=5n%5E2-11n ALIGN=MIDDLE  ALT=\"S%5Bn%5D=5n%5E2-11n\"   DISABLED_style= \"max-width:90%; height:auto;\" > :<BR>\n" );
document.write( "The sum of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n-1 ALIGN=MIDDLE  ALT=\"n-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > terms is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S%5Bn-1%5D=5%28n-1%29%5E2-11%28n-1%29 ALIGN=MIDDLE  ALT=\"S%5Bn-1%5D=5%28n-1%29%5E2-11%28n-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S%5Bn-1%5D=5%28n%5E2-2n%2B1%29-11n%2B11 ALIGN=MIDDLE  ALT=\"S%5Bn-1%5D=5%28n%5E2-2n%2B1%29-11n%2B11\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S%5Bn-1%5D=5n%5E2-10n%2B5-11n%2B11 ALIGN=MIDDLE  ALT=\"S%5Bn-1%5D=5n%5E2-10n%2B5-11n%2B11\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S%5Bn-1%5D=5n%5E2-21n%2B16 ALIGN=MIDDLE  ALT=\"S%5Bn-1%5D=5n%5E2-21n%2B16\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "and since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S%5Bn%5D=S%5Bn-1%5D%2Ba%5Bn%5D ALIGN=MIDDLE  ALT=\"S%5Bn%5D=S%5Bn-1%5D%2Ba%5Bn%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5Bn%5D ALIGN=MIDDLE  ALT=\"a%5Bn%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > is 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( "we can find an expression for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5Bn%5D=S%5Bn%5D-S%5Bn-1%5D ALIGN=MIDDLE  ALT=\"a%5Bn%5D=S%5Bn%5D-S%5Bn-1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5Bn%5D=5n%5E2-11n-%285n%5E2-21n%2B16%29 ALIGN=MIDDLE  ALT=\"a%5Bn%5D=5n%5E2-11n-%285n%5E2-21n%2B16%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5Bn%5D=5n%5E2-11n-5n%5E2%2B21n-16%29 ALIGN=MIDDLE  ALT=\"a%5Bn%5D=5n%5E2-11n-5n%5E2%2B21n-16%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5Bn%5D=10n-16%29 ALIGN=MIDDLE  ALT=\"a%5Bn%5D=10n-16%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5Bn%5D=10n-10-6%29 ALIGN=MIDDLE  ALT=\"a%5Bn%5D=10n-10-6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5Bn%5D=10%28n-1%29-6%29 ALIGN=MIDDLE  ALT=\"a%5Bn%5D=10%28n-1%29-6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "That gives us <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28d=10%29 ALIGN=MIDDLE  ALT=\"highlight%28d=10%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > as the common difference,<BR>\n" );
document.write( "and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B1%5D=-6 ALIGN=MIDDLE  ALT=\"a%5B1%5D=-6\"   DISABLED_style= \"max-width:90%; height:auto;\" > as the first term.<BR>\n" );
document.write( "We know that in an arithmetic sequence<BR>\n" );
document.write( "with a first term <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B1%5D ALIGN=MIDDLE  ALT=\"a%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and 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;\" > ,<BR>\n" );
document.write( "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%5B1%5D%2Bd%28n-1%29 ALIGN=MIDDLE  ALT=\"a%5Bn%5D=a%5B1%5D%2Bd%28n-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d ALIGN=MIDDLE  ALT=\"d\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the coefficient of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28n-1%29 ALIGN=MIDDLE  ALT=\"%28n-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and the other term is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B1%5D ALIGN=MIDDLE  ALT=\"a%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "WHY WE KNOW <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S%5Bn%5D=5n%5E2-11 ALIGN=MIDDLE  ALT=\"S%5Bn%5D=5n%5E2-11\"   DISABLED_style= \"max-width:90%; height:auto;\" > IS WRONG:<BR>\n" );
document.write( "We have the \"formula\" <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S%5Bn%5D=%282a%5B1%5D%2B%28n-1%29d%29n%2F2 ALIGN=MIDDLE  ALT=\"S%5Bn%5D=%282a%5B1%5D%2B%28n-1%29d%29n%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "In that expression <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 a factor.<BR>\n" );
document.write( "The expression is a quadratic (degree 2) polynomial with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%5D%7D%7D+as+a+factor%2C%0D%0Ameaning+no+independent+term.%0D%0A%7B%7B%7BS%5Bn%5D=%282a%5B1%5D%2Bnd-d%29n%2F2 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S%5Bn%5D=%28%282a%5B1%5D-d%29%2Bnd%29n%2F2 ALIGN=MIDDLE  ALT=\"S%5Bn%5D=%28%282a%5B1%5D-d%29%2Bnd%29n%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S%5Bn%5D=%28dn%5E2%2B%282a%5B1%5D-d%29n%29%2F2 ALIGN=MIDDLE  ALT=\"S%5Bn%5D=%28dn%5E2%2B%282a%5B1%5D-d%29n%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S%5Bn%5D=%28d%2F2%29n%5E2%2B%28%282a%5B1%5D-d%29%2F2%29n ALIGN=MIDDLE  ALT=\"S%5Bn%5D=%28d%2F2%29n%5E2%2B%28%282a%5B1%5D-d%29%2F2%29n\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );