document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1103117>Question 1103117</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the sum of the first 24 terms of the arithmetic series.<BR>\n" );
document.write( "1+8+15+22+...<BR>\n" );
document.write( "S(24)= </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #717792 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=717792\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "In ANY set of numbers, the sum of all the numbers is the average, multiplied by how many numbers there are.<br><BR>\n" );
document.write( "In an arithmetic series, since the terms are \"evenly spaced\", the average of all the numbers is the average of the first and last terms.<br><BR>\n" );
document.write( "So the sum of 24 terms of an arithmetic series is the average of the first and last terms, multiplied by 24.<br><BR>\n" );
document.write( "The 24th term is the first term, plus the common difference added 23 times.<br><BR>\n" );
document.write( "For this example, then...<BR>\n" );
document.write( "The 24th term is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2B23%2A7+=+1%2B161+=+162 ALIGN=MIDDLE  ALT=\"1%2B23%2A7+=+1%2B161+=+162\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The average of the first and last terms is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2B162%29%2F2+=+81.5 ALIGN=MIDDLE  ALT=\"%281%2B162%29%2F2+=+81.5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "And the sum of the first 24 terms is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=81.5%2A24+=+1956 ALIGN=MIDDLE  ALT=\"81.5%2A24+=+1956\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Note that many students prefer a different way of thinking of the sum of the terms of an arithmetic series.  Instead of thinking <BR>\n" );
document.write( "\"(average of first and last) times (the number of terms)\",<BR>\n" );
document.write( "They group the numbers in pairs (1st term with 24th, 2nd with 23rd, and so on) so that each pair has the same sum; then they get the sum of all the terms in the series as<BR>\n" );
document.write( "\"(sum of first and last terms) times (the number of pairs)\".<br><BR>\n" );
document.write( "For your example, that method of thinking of the sum would give you...<BR>\n" );
document.write( "There are 24 terms; so that is 12 pairs<BR>\n" );
document.write( "The sum of the first and last terms is 1+162=163;<BR>\n" );
document.write( "The sum of all the terms is 163*12 = 1956. </SPAN>\n" );
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