document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=524900>Question 524900</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The first three terms of an arithmetic progression are tan x, cos x, and sec x, respectively. If the k th term is cot x, find k. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #348896 by </B><A HREF=/tutors/aboutme.mpl?userid=Aswathy><B>Aswathy(23)</B></A><img src=\"/images/stars/stars-07.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Aswathy><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=348896\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Note that AP means arithmetic progression.<BR>\n" );
document.write( "Here, a1=tanx, a2=cosx and a3=secx<BR>\n" );
document.write( "Common difference (d)of the given AP=a2-a1=cosx-tanx=cosx-(sinx/cosx)<BR>\n" );
document.write( "                                 =(cos^2x-sinx)/(cosx)..............(1)<BR>\n" );
document.write( "Similarly,a3-a2=secx-cosx=(1/cosx)-cosx=(1-cos^2x)/cosx=(sin^2x)/cosx........(2)<BR>\n" );
document.write( "Since the given terms are in AP,then the difference between the terms will be equal.<BR>\n" );
document.write( "Therefore,a2-a1=a3-a2<BR>\n" );
document.write( "then,cosx-tanx=secx-cosx<BR>\n" );
document.write( "cosx+cosx=secx+tanx<BR>\n" );
document.write( "2cosx=(1/cosx)+(sinx/cosx)<BR>\n" );
document.write( "2cosx=(1+sinx)/cosx<BR>\n" );
document.write( "2cos^2x=1+sinx.................(3)<BR>\n" );
document.write( "Similarly,now from (1) and(2),<BR>\n" );
document.write( "(cos^2x-sinx)/(cosx)=(sin^2x)/(cosx)<BR>\n" );
document.write( "then,cos^2x-sinx=sin^2x  {cancelling out cosx on both sides of the equation}<BR>\n" );
document.write( "cos^2x=sinx+sin^2x<BR>\n" );
document.write( "cos^2x=sinx(1+sinx)<BR>\n" );
document.write( "cos^2x=sinx(2cos^2x)  [from (3)]<BR>\n" );
document.write( "cos^2x/2cos^2x=sinx<BR>\n" );
document.write( "1/2=sinx {cancelling out cos^2x on both sides of the equation}<BR>\n" );
document.write( "Therefore,x=30 as sinx=1/2<BR>\n" );
document.write( "Let ak(k is subscript of a) be the kth term which is cotx here.<BR>\n" );
document.write( "In the given AP,d=a2-a1=cos30-tan30=1/2sqrt(3)<BR>\n" );
document.write( "Now,kth term ,ak=a1+(k-1)d<BR>\n" );
document.write( "then,cos30=tan30+(k-1)2/2sqrt(3)\r<BR>\n" );
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document.write( "By solving this equation we get,k=5 which is our required answer.<BR>\n" );
document.write( "So,k=5\r<BR>\n" );
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document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );