document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=251659>Question 251659</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Q1 If a,b,c,d are in Harmonic progression show that(a-c)(b-d)=4(a-b)(c-d)<BR>\n" );
document.write( "Q2 If a+b,b+c,c+a are in H.P. show that a^2,b^2,c^2 are in A.P.<BR>\n" );
document.write( "Q3 If x,y,z are in A.P.x,xy,z are in G.P.show thatx,x^2y,z are in H.P. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #183863 by </B><A HREF=/tutors/aboutme.mpl?userid=palanisamy><B>palanisamy(496)</B></A><img src=\"/images/stars/stars-10.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=palanisamy><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=183863\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Q1.<BR>\n" );
document.write( "    a,b,c,d are in H.P<BR>\n" );
document.write( "  Then 1/a, 1/b, 1/c, 1/d will be in A.P<BR>\n" );
document.write( "Therefore  2(1/b-1/a) = (1/c-1/a)  and 2(1/d-1/c) = (1/d-1/b) <BR>\n" );
document.write( "           2(a-b)/ba  = (a-c)/ca   and 2(c-d)/dc  = (b-d)/db<BR>\n" );
document.write( "Multiplying these two equations, we get<BR>\n" );
document.write( "          4(a-b)(c-d)/abcd = (a-c)(b-d)/abcd<BR>\n" );
document.write( "          4(a-b)(c-d) = (a-c)(b-d)\r<BR>\n" );
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document.write( "Q2.  <BR>\n" );
document.write( " Given a+b,b+c,c+a are in H.P.<BR>\n" );
document.write( "Therefore, 1/(a+b) + 1/(c+a)      = 2/(b+c)<BR>\n" );
document.write( "             (c+a+a+b)/(a+b)(c+a) = 2/(b+c)<BR>\n" );
document.write( "             (c+2a+b)(b+c)        = 2(a+b)(c+a)<BR>\n" );
document.write( "      bc+c^2+2ab+2ac+b^2+bc       = 2ac+2a^2+2bc+2ab<BR>\n" );
document.write( "                 c^2+b^2          = 2a^2<BR>\n" );
document.write( "                   (b^2+c^2)/2    =a^2<BR>\n" );
document.write( "Therefore b^2,a^2,c^2 are in A.P\r<BR>\n" );
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document.write( "Q3 Given, x,y,z are in A.P. <BR>\n" );
document.write( "Therefore, x+z = 2y  ......(1)<BR>\n" );
document.write( "Also, x,xy,z are in G.P<BR>\n" );
document.write( "Therefore, (xy)^2 = xz<BR>\n" );
document.write( "           x^2y^2 = xz<BR>\n" );
document.write( "             xy^2 = z  ....(2)<BR>\n" );
document.write( "Next, we will prove x,x^2y,z are in H.P<BR>\n" );
document.write( "Now, 1/x+1/z = (z+x)/xz<BR>\n" );
document.write( "             = 2y/x.xy^2<BR>\n" );
document.write( "             = 2/x^2y<BR>\n" );
document.write( "Therefore, 1/x+1/z =2/x^2y<BR>\n" );
document.write( "Therefore,x,x^2y,z are in H.P\r<BR>\n" );
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document.write( "       </SPAN>\n" );
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