document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1182882>Question 1182882</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The sum of the first ten terms of an arithmetic progression consisting of positive integer terms is equal to the sum of the 20th, 21st and 22nd term. If the first term is less than 20, find how many terms are required to give a sum of 960. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #812986 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=812986\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "The sum of the first n terms of an arithmetic progression with first term a and constant difference d is<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S%28n%29=%28n%282a%2B%28n-1%29d%29%29%2F2 ALIGN=MIDDLE  ALT=\"S%28n%29=%28n%282a%2B%28n-1%29d%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "We will use that formula twice in solving this problem.<br><BR>\n" );
document.write( "The sum of the first ten terms is<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S%2810%29+=+%2810%282a%2B9d%29%29%2F2 ALIGN=MIDDLE  ALT=\"S%2810%29+=+%2810%282a%2B9d%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "The sum of the 20th, 21st, and 22nd terms is three times the 21st term:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%28a%2B20d%29 ALIGN=MIDDLE  ALT=\"3%28a%2B20d%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "The problem then tells us<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2810%282a%2B9d%29%2F2%29=3%28a%2B20d%29 ALIGN=MIDDLE  ALT=\"%2810%282a%2B9d%29%2F2%29=3%28a%2B20d%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10a%2B45d=3a%2B60d ALIGN=MIDDLE  ALT=\"10a%2B45d=3a%2B60d\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7a=15d ALIGN=MIDDLE  ALT=\"7a=15d\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "The sequence contains only positive integers; and the first term a is less than 20.  Since 7 and 15 have no common factor, the only solution that meets the requirements is a=15 and d=7.  So the progression is<br><BR>\n" );
document.write( "15, 22, 29, 36, ...<br><BR>\n" );
document.write( "The problem says to find the number of terms n for which the sum is 960:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S%28n%29+=+%28n%2830%2B%28n-1%297%29%29%2F2+=+960 ALIGN=MIDDLE  ALT=\"S%28n%29+=+%28n%2830%2B%28n-1%297%29%29%2F2+=+960\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=30n%2B7n%5E2-7n+=+1920 ALIGN=MIDDLE  ALT=\"30n%2B7n%5E2-7n+=+1920\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7n%5E2%2B23n-1920+=+0 ALIGN=MIDDLE  ALT=\"7n%5E2%2B23n-1920+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Since that won't factor easily, use the quadratic formula to find n=15.<br><BR>\n" );
document.write( "ANSWER: 15 terms of the sequence will have a sum of 960.<br><BR>\n" );
document.write( "CHECK:<BR>\n" );
document.write( "1st term 15<BR>\n" );
document.write( "10th term 15+9(7)=78<BR>\n" );
document.write( "21st term 15+20(7)=155<br><BR>\n" );
document.write( "sum of first 10 terms: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2810%2815%2B78%29%29%2F2=5%2893%29=465 ALIGN=MIDDLE  ALT=\"%2810%2815%2B78%29%29%2F2=5%2893%29=465\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "sum of 20th, 21st, and 22nd terms: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%28155%29=465 ALIGN=MIDDLE  ALT=\"3%28155%29=465\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );