document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1124987>Question 1124987</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The first and second terms of an arithmetic sequence are a and b respectively. If the nth term is c, express n in terms of a b and c and hence find the sum of these n terms. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #741421 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=741421\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "The first term is a; the second is b; so the common difference is (b-a).<br><BR>\n" );
document.write( "The n-th term is then the first term plus the common difference (n-1) times:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c+=+a+%2B+%28n-1%29%28b-a%29 ALIGN=MIDDLE  ALT=\"c+=+a+%2B+%28n-1%29%28b-a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c+=+a+%2B+nb+-+na+-+b+%2B+a ALIGN=MIDDLE  ALT=\"c+=+a+%2B+nb+-+na+-+b+%2B+a\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c+=+2a-b%2Bnb-na ALIGN=MIDDLE  ALT=\"c+=+2a-b%2Bnb-na\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%2Bc-2a+=+n%28b-a%29 ALIGN=MIDDLE  ALT=\"b%2Bc-2a+=+n%28b-a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n+=+%28b%2Bc-2a%29%2F%28b-a%29 ALIGN=MIDDLE  ALT=\"n+=+%28b%2Bc-2a%29%2F%28b-a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >  <-- answer to the first question<br><BR>\n" );
document.write( "The sum of the series is (number of terms) times (average of first and last terms):<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S+=+n%28%28a%2Bc%29%2F2%29 ALIGN=MIDDLE  ALT=\"S+=+n%28%28a%2Bc%29%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S+=+%28%28b%2Bc-2a%29%2F%28b-a%29%29%28%28a%2B%282a-b%2Bnb-na%29%29%2F2%29 ALIGN=MIDDLE  ALT=\"S+=+%28%28b%2Bc-2a%29%2F%28b-a%29%29%28%28a%2B%282a-b%2Bnb-na%29%29%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S+=+%28%28b%2Bc-2a%29%283a-b%2Bnb-na%29%29%2F%282%28b-a%29%29 ALIGN=MIDDLE  ALT=\"S+=+%28%28b%2Bc-2a%29%283a-b%2Bnb-na%29%29%2F%282%28b-a%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >  <-- answer to the second question<br><BR>\n" );
document.write( "Let's check these formulas with an example -- just in case we made an error in our algebra....<br><BR>\n" );
document.write( "a = 3; b = 5; 9 terms.<br><BR>\n" );
document.write( "The common difference is b-a = 2; the 9th term, c, is 3+8(2) = 19.<br><BR>\n" );
document.write( "We know the number of terms is 9; our formula says the number of terms  should be<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28b%2Bc-2a%29%2F%28b-a%29 ALIGN=MIDDLE  ALT=\"%28b%2Bc-2a%29%2F%28b-a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%285%2B19-2%283%29%29%2F%285-3%29+=+18%2F2+=+9 ALIGN=MIDDLE  ALT=\"%285%2B19-2%283%29%29%2F%285-3%29+=+18%2F2+=+9\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Our formula gives the right result for this example.<br><BR>\n" );
document.write( "The sum of the series is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9%28%283%2B19%29%2F2%29+=+9%2A11+=+99 ALIGN=MIDDLE  ALT=\"9%28%283%2B19%29%2F2%29+=+9%2A11+=+99\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<br><BR>\n" );
document.write( "Our formula says the sum is supposed to be<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28b%2Bc-2a%29%283a-b%2Bnb-na%29%29%2F%282%28b-a%29%29 ALIGN=MIDDLE  ALT=\"%28%28b%2Bc-2a%29%283a-b%2Bnb-na%29%29%2F%282%28b-a%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%285%2B19-2%283%29%29%283%283%29-5%2B9%285%29-9%283%29%29%29%2F%282%285-3%29%29 ALIGN=MIDDLE  ALT=\"%28%285%2B19-2%283%29%29%283%283%29-5%2B9%285%29-9%283%29%29%29%2F%282%285-3%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%2824-6%29%289-5%2B45-27%29%29%2F4+=+%2818%2A22%29%2F4+=+9%2A11+=+99 ALIGN=MIDDLE  ALT=\"%28%2824-6%29%289-5%2B45-27%29%29%2F4+=+%2818%2A22%29%2F4+=+9%2A11+=+99\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Our formula for the sum also gives the right result for this example. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );