document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=260663>Question 260663</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A sequence begins with 1, 4, 7, … Another sequence begins with<BR>\n" );
document.write( "99, 95, 91, … At which position is the positive difference between<BR>\n" );
document.write( "the respective terms of the two sequences the minimum?<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #192015 by </B><A HREF=/tutors/aboutme.mpl?userid=jsmallt9><B>jsmallt9(3759)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jsmallt9><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=192015\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Both of these sequences are arithmetic sequences. The general formula for the terms of an arithmetic sequence is:<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%29d ALIGN=MIDDLE  ALT=\"a%5Bn%5D+=+a%5B1%5D+%2B+%28n-1%29d\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "For the first sequence, since the first term, <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;\" >, is 1 and the common difference, d, is 3 we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5Bn%5D+=+1+%2B+%28n-1%293 ALIGN=MIDDLE  ALT=\"a%5Bn%5D+=+1+%2B+%28n-1%293\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "For the second sequence we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5Bn%5D+=+99+%2B+%28n-1%29%28-4%29 ALIGN=MIDDLE  ALT=\"a%5Bn%5D+=+99+%2B+%28n-1%29%28-4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The positive difference in these terms, which we'll call D, is:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D+=+abs%28%281+%2B+%28n-1%293%29+-+%2899+%2B+%28n-1%29%28-4%29%29%29 ALIGN=MIDDLE  ALT=\"D+=+abs%28%281+%2B+%28n-1%293%29+-+%2899+%2B+%28n-1%29%28-4%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "(Note how we use absolute value to ensure a positive difference.) Simplifying this we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D+=+abs%28%281+%2B+3n-3%29+-+%2899+%2B+-4n+%2B+4%29%29 ALIGN=MIDDLE  ALT=\"D+=+abs%28%281+%2B+3n-3%29+-+%2899+%2B+-4n+%2B+4%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=D+=+abs%28%283n-2%29+-+%28103+%2B+-4n%29%29 ALIGN=MIDDLE  ALT=\"D+=+abs%28%283n-2%29+-+%28103+%2B+-4n%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=D+=+abs%287n+-+105%29 ALIGN=MIDDLE  ALT=\"D+=+abs%287n+-+105%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So the question is: What value of n makes 7n - 105 closest to zero? After a little effort we find that if n = 15, then 7n - 105 is zero! In other words, the 15th term of both sequences is the same!\r<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );