document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1184224>Question 1184224</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The coefficients of the powers of x in the 2nd, 3rd and 4th term of the expansion <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%281+%2B+x%29%5En+ ALIGN=MIDDLE  ALT=\"+%281+%2B+x%29%5En+\"   DISABLED_style= \"max-width:90%; height:auto;\" > is in arithmetic progression where n is positive integer. Find the number of coefficients of the odd powers of x in the expansion?  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #814787 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13203)</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=814787\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "The coefficients that are in arithmetic progression can be found algebraically; however, it is unclear what the question is asking for....<br><BR>\n" );
document.write( "The coefficients in the expansion of (1+x)^n are...<br><BR>\n" );
document.write( "1st term: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "2nd term: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n ALIGN=MIDDLE  ALT=\"n\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "3rd term: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28n%29%28n-1%29%29%2F2 ALIGN=MIDDLE  ALT=\"%28%28n%29%28n-1%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "4th term: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28n%29%28n-1%29%28n-2%29%29%2F6 ALIGN=MIDDLE  ALT=\"%28%28n%29%28n-1%29%28n-2%29%29%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "The 2nd, 3rd, and 4th terms are in arithmetic progression, so the 3rd term is twice the sum of the 2nd and 4th terms:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%28%28n%29%28n-1%29%29%2F2=n%2B%28%28n%29%28n-1%29%28n-2%29%29%2F6 ALIGN=MIDDLE  ALT=\"2%28%28n%29%28n-1%29%29%2F2=n%2B%28%28n%29%28n-1%29%28n-2%29%29%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%28n-1%29=n%281%2B%28%28n-1%29%28n-2%29%29%2F6%29 ALIGN=MIDDLE  ALT=\"n%28n-1%29=n%281%2B%28%28n-1%29%28n-2%29%29%2F6%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-1=1%2B%28n%5E2-3n%2B2%29%2F6 ALIGN=MIDDLE  ALT=\"n-1=1%2B%28n%5E2-3n%2B2%29%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n-2=%28n%5E2-3n%2B2%29%2F6 ALIGN=MIDDLE  ALT=\"n-2=%28n%5E2-3n%2B2%29%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6n-12=n%5E2-3n%2B2%29 ALIGN=MIDDLE  ALT=\"6n-12=n%5E2-3n%2B2%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%5E2-9n-14=0 ALIGN=MIDDLE  ALT=\"n%5E2-9n-14=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-2%29=0 ALIGN=MIDDLE  ALT=\"%28n-7%29%28n-2%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "n=7 or n=2<br><BR>\n" );
document.write( "n=2 makes no sense in the problem because (1+x)^2 only has 3 terms.<br><BR>\n" );
document.write( "So n=7.<br><BR>\n" );
document.write( "The 2nd, 3rd, and 4th terms in the expansion of (1+x)^7 are 7, 21, and 35 -- in arithmetic progression with common difference 14.<br><BR>\n" );
document.write( "But I don't know how to answer the question because I don't know what it means:<br><BR>\n" );
document.write( "\"Find the number of coefficients of the odd powers of x in the expansion.\"<br><BR>\n" );
document.write( "The expansion of (1+x)^7 has 8 terms; half of them are even powers of x and half are odd powers.<br><BR>\n" );
document.write( "So I guess the answer is 4.<br><BR>\n" );
document.write( "But after doing some good math to find a value of n for which the coefficients of the 2nd, 3rd, and 4th terms in the expansion of (1+x)^n form an arithmetic progression, that seems like a very odd question to ask....<br><BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );