document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1094750>Question 1094750</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The sum of the first twenty terms in an arithmetic progression is 840 and the common difference is 4.Find<BR>\n" );
document.write( "(a)the first term<BR>\n" );
document.write( "(b)the sum of the first ten terms </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #709339 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=709339\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "A key concept for working with arithmetic progressions is that we can group the terms in pairs, with each pair having the same sum.<br><BR>\n" );
document.write( "In your example, where the sum of the first 20 terms is 840, we know those 20 terms are 10 pairs, so the sum of each pair is 840/10 = 84.<br><BR>\n" );
document.write( "One of the pairs is the first and last (20th) terms.  The 20th term is the first term, plus the common difference (which is 4) 19 times; and the sum of the first and 20th terms is 84.  So<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+%2B+%28a%2B19%284%29%29+=+2a%2B76+=+84 ALIGN=MIDDLE  ALT=\"a+%2B+%28a%2B19%284%29%29+=+2a%2B76+=+84\"   DISABLED_style= \"max-width:90%; height:auto;\" >  -->  a = 4.<br><BR>\n" );
document.write( "That's the answer to part (a).<br><BR>\n" );
document.write( "Part (b): In finding the sum of the first 10 terms, we will have 5 pairs with the same sum; one of those pairs is the 1st and 10th terms.  The 10th term is the first term 4, plus the common difference (4) 9 times; 4+9(4) = 40.  So the sum of each of the 5 pairs is 4+40 = 44; since there are 5 pairs, the sum of the first 10 terms is 44*5 = 220. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );