document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=855491>Question 855491</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The sixth term of an AP is 37 an the tenth term 53.Find the sum of the first twenty terms. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #515395 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=515395\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> MENTAL MATH (for multiple choice tests):<BR>\n" );
document.write( "from one number to the next you are adding a common difference.<BR>\n" );
document.write( "From the sixth term to the tenth term that common difference was added 4 times because<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10-6=4 ALIGN=MIDDLE  ALT=\"10-6=4\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "That means that the common difference is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2853-37%29%2F4=16%2F4=highlight%284%29 ALIGN=MIDDLE  ALT=\"%2853-37%29%2F4=16%2F4=highlight%284%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "In an AP, the 1st and 20th terms<BR>\n" );
document.write( "add to the same sum as<BR>\n" );
document.write( "the 2nd and 19th terms,<BR>\n" );
document.write( "the 3rd and 18th terms, and so on until we get to<BR>\n" );
document.write( "the 10th and 11th terms, in the middle of the first 20 terms.<BR>\n" );
document.write( "The 10th term is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=53 ALIGN=MIDDLE  ALT=\"53\"   DISABLED_style= \"max-width:90%; height:auto;\" > as we were told,<BR>\n" );
document.write( "and the 11th term is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=53%2B4=57 ALIGN=MIDDLE  ALT=\"53%2B4=57\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "The sum of 20 first terms is made of the 10 sums listed above,<BR>\n" );
document.write( "and each of those sums is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=53%2B57=110 ALIGN=MIDDLE  ALT=\"53%2B57=110\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "So the sum of the first 20 terms is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10%2A110=highlight%281100%29 ALIGN=MIDDLE  ALT=\"10%2A110=highlight%281100%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "That is a lot of words, but you can think through the concepts and calculations much faster than you can explain and write them.<BR>\n" );
document.write( " <BR>\n" );
document.write( "USING FORMULAS (to please the teacher by \"showing your work\"):<BR>\n" );
document.write( "<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;\" >= first term<BR>\n" );
document.write( "<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=a%5Bn%5D=a%5B1%5D%2B%28n-1%29%2Ad ALIGN=MIDDLE  ALT=\"a%5Bn%5D=a%5B1%5D%2B%28n-1%29%2Ad\"   DISABLED_style= \"max-width:90%; height:auto;\" >= general formula for 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( "For <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n=6 ALIGN=MIDDLE  ALT=\"n=6\"   DISABLED_style= \"max-width:90%; height:auto;\" > we know that<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B6%5D=a%5B1%5D%2B%286-1%29%2Ad=a%5B1%5D%2B5%2Ad=37 ALIGN=MIDDLE  ALT=\"a%5B6%5D=a%5B1%5D%2B%286-1%29%2Ad=a%5B1%5D%2B5%2Ad=37\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "For <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;\" > we know that<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B10%5D=a%5B1%5D%2B%2810-1%29%2Ad=a%5B1%5D%2B9%2Ad=53 ALIGN=MIDDLE  ALT=\"a%5B10%5D=a%5B1%5D%2B%2810-1%29%2Ad=a%5B1%5D%2B9%2Ad=53\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28a%5B1%5D%2B5%2Ad=37%2Ca%5B1%5D%2B9%2Ad=53%29 ALIGN=MIDDLE  ALT=\"system%28a%5B1%5D%2B5%2Ad=37%2Ca%5B1%5D%2B9%2Ad=53%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is solved to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28system%28a%5B1%5D=17%2Cd=4%29%29 ALIGN=MIDDLE  ALT=\"highlight%28system%28a%5B1%5D=17%2Cd=4%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Then we apply some formula for the sum of 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.<BR>\n" );
document.write( "Maybe the formula you were given is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S%5Bn%5D=%282%2Aa%5B1%5D%2B%28n-1%29%2Ad%29%2A%28n%2F2%29 ALIGN=MIDDLE  ALT=\"S%5Bn%5D=%282%2Aa%5B1%5D%2B%28n-1%29%2Ad%29%2A%28n%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > (or some variation of that).<BR>\n" );
document.write( "Or maybe you were given<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S%5Bn%5D=%28a%5B1%5D%2Ba%5Bn%5D%29%2An%2F2 ALIGN=MIDDLE  ALT=\"S%5Bn%5D=%28a%5B1%5D%2Ba%5Bn%5D%29%2An%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > (or some variation of that).<BR>\n" );
document.write( "With <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S%5Bn%5D=%28a%5B1%5D%2Ba%5Bn%5D%29%2An%2F2 ALIGN=MIDDLE  ALT=\"S%5Bn%5D=%28a%5B1%5D%2Ba%5Bn%5D%29%2An%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5Bn%5D=a%5B1%5D%2B%28n-1%29%2Ad ALIGN=MIDDLE  ALT=\"a%5Bn%5D=a%5B1%5D%2B%28n-1%29%2Ad\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "you would calculate the 20th term as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B20%5D=17%2B%2820-1%29%2A4=17%2B19%2A4=17%2B76=93 ALIGN=MIDDLE  ALT=\"a%5B20%5D=17%2B%2820-1%29%2A4=17%2B19%2A4=17%2B76=93\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and then you would calculate the sum of the first 20 terms as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S%5B20%5D=%2817%2B93%29%2A20%2F2=110%2A20%2F2=highlight%281100%29 ALIGN=MIDDLE  ALT=\"S%5B20%5D=%2817%2B93%29%2A20%2F2=110%2A20%2F2=highlight%281100%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "With <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S%5Bn%5D=%282%2Aa%5B1%5D%2B%28n-1%29%2Ad%29%2A%28n%2F2%29 ALIGN=MIDDLE  ALT=\"S%5Bn%5D=%282%2Aa%5B1%5D%2B%28n-1%29%2Ad%29%2A%28n%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > you would calculate<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S%5B20%5D=%282%2A17%2B%2820-1%29%2A4%29%2A%2820%2F2%29=%2834%2B19%2A4%29%2A10=%2834%2B76%29%2A10=110%2A10=highlight%281100%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );