document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1117570>Question 1117570</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In the expansion of (1+x)^n,  the coefficient of x^5 is the arithmetic mean of the coefficients of x^4 and x^6. calculate the possible values of n </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #732684 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13215)</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=732684\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "The coefficients are....<br><BR>\n" );
document.write( "C(n,4) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28n%28n-1%29%28n-2%29%28n-3%29%29%2F24 ALIGN=MIDDLE  ALT=\"%28n%28n-1%29%28n-2%29%28n-3%29%29%2F24\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "C(n,5) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28n%28n-1%29%28n-2%29%28n-3%29%28n-4%29%29%2F120 ALIGN=MIDDLE  ALT=\"%28n%28n-1%29%28n-2%29%28n-3%29%28n-4%29%29%2F120\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "C(n,6) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28n%28n-1%29%28n-2%29%28n-3%29%28n-4%29%28n-5%29%29%2F720 ALIGN=MIDDLE  ALT=\"%28n%28n-1%29%28n-2%29%28n-3%29%28n-4%29%28n-5%29%29%2F720\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "We need to find the value(s) of n for which C(n,5) is the arithmetic mean of C(n,4) and C(n,6).<br><BR>\n" );
document.write( "An interesting problem; but the algebra works out relatively easily....<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28n%28n-1%29%28n-2%29%28n-3%29%28n-4%29%29%2F120+=+%28%28n%28n-1%29%28n-2%29%28n-3%29%29%2F24%2B%28n%28n-1%29%28n-2%29%28n-3%29%28n-4%29%28n-5%29%29%2F720%29%2F2+=+%28n%28n-1%29%28n-2%29%28n-3%29%29%2F48%2B%28n%28n-1%29%28n-2%29%28n-3%29%28n-4%29%28n-5%29%29%2F1440%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Multiply by the common denominator 1440 and cancel the common factors n through n-3:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12%28n-4%29+=+30%2B%28n-4%29%28n-5%29 ALIGN=MIDDLE  ALT=\"12%28n-4%29+=+30%2B%28n-4%29%28n-5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12n-48+=+30%2Bn%5E2-9n%2B20 ALIGN=MIDDLE  ALT=\"12n-48+=+30%2Bn%5E2-9n%2B20\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%5E2-21n%2B98+=+0 ALIGN=MIDDLE  ALT=\"n%5E2-21n%2B98+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28n-7%29%28n-14%29+=+0 ALIGN=MIDDLE  ALT=\"%28n-7%29%28n-14%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "The two solutions are n=7 and n=14.<br><BR>\n" );
document.write( "Check:<br><BR>\n" );
document.write( "For n=7, the coefficients are 7, 21, and 35; 21 = (7+35)/2.<br><BR>\n" );
document.write( "For n=14, the coefficients are 1001, 2002, and 3003; 2002 = (1001+3003)/2.<br><BR>\n" );
document.write( "DONE!<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );