document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=928076>Question 928076</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In an arithmetic series, t10=21 and t19=48    \r<BR>\n" );
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document.write( "Find a, d, and S23\r<BR>\n" );
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document.write( "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 #563617 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=563617\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> I assume <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" >= first term, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t%5B10%5D ALIGN=MIDDLE  ALT=\"t%5B10%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >= tenth term, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t%5B19%5D ALIGN=MIDDLE  ALT=\"t%5B19%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >= term number <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=19 ALIGN=MIDDLE  ALT=\"19\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S%5B23%5D ALIGN=MIDDLE  ALT=\"S%5B23%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >= sum of first <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=23 ALIGN=MIDDLE  ALT=\"23\"   DISABLED_style= \"max-width:90%; height:auto;\" > terms.<BR>\n" );
document.write( "It is commonly understood that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d ALIGN=MIDDLE  ALT=\"d\"   DISABLED_style= \"max-width:90%; height:auto;\" >= common difference.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t%5Bn%5D=a%2B%28n-1%29d ALIGN=MIDDLE  ALT=\"t%5Bn%5D=a%2B%28n-1%29d\"   DISABLED_style= \"max-width:90%; height:auto;\" > , so<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%2821=a%2B9d%2C48=a%2B18d%29 ALIGN=MIDDLE  ALT=\"system%2821=a%2B9d%2C48=a%2B18d%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=48-21=18d-9d ALIGN=MIDDLE  ALT=\"48-21=18d-9d\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=27=9d ALIGN=MIDDLE  ALT=\"27=9d\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=27%2F3=d ALIGN=MIDDLE  ALT=\"27%2F3=d\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28d=3%29 ALIGN=MIDDLE  ALT=\"highlight%28d=3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Then, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%2821=a%2B9d%2Cd=3%29 ALIGN=MIDDLE  ALT=\"system%2821=a%2B9d%2Cd=3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=21=a%2B9%2A3 ALIGN=MIDDLE  ALT=\"21=a%2B9%2A3\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=21=a%2B27 ALIGN=MIDDLE  ALT=\"21=a%2B27\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=21-27=a ALIGN=MIDDLE  ALT=\"21-27=a\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28a=-6%29 ALIGN=MIDDLE  ALT=\"highlight%28a=-6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "There are different formulas to calculate sums of arithmetic sequences, but I prefer to remember that adding up the first <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n ALIGN=MIDDLE  ALT=\"n\"   DISABLED_style= \"max-width:90%; height:auto;\" > terms twice, grouping them in pairs \"head-to-tail\", <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t%5B0%5D%2Bt%5Bn%5D ALIGN=MIDDLE  ALT=\"t%5B0%5D%2Bt%5Bn%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t%5B1%5D%2Bt%5Bn-1%5D ALIGN=MIDDLE  ALT=\"t%5B1%5D%2Bt%5Bn-1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t%5B2%5D%2Bt%5Bn-2%5D ALIGN=MIDDLE  ALT=\"t%5B2%5D%2Bt%5Bn-2%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and so on, you end up with a sum 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;\" > pairs that all add up to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t%5B0%5D%2Bt%5Bn%5D ALIGN=MIDDLE  ALT=\"t%5B0%5D%2Bt%5Bn%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2S%5Bn%5D=n%28t%5B0%5D%2Bt%5Bn%5D%29 ALIGN=MIDDLE  ALT=\"2S%5Bn%5D=n%28t%5B0%5D%2Bt%5Bn%5D%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S%5Bn%5D=%28n%28t%5B0%5D%2Bt%5Bn%5D%29%29%2F2 ALIGN=MIDDLE  ALT=\"S%5Bn%5D=%28n%28t%5B0%5D%2Bt%5Bn%5D%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "So, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t%5B23%5D=-6%2B22%2A3=-6%2B66=60 ALIGN=MIDDLE  ALT=\"t%5B23%5D=-6%2B22%2A3=-6%2B66=60\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S%5B23%5D=%2823%28a%2Bt%5B23%5D%29%29%2F2=%2823%28-6%2B60%29%29%2F2=%2823%2854%29%29%2F2=highlight%28621%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > . </SPAN>\n" );
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