document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=26646>Question 26646</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Let (Fn)=(1,1,2,3,5,8,13,21,34,55,...) be the fibonacci sequence defined by <BR>\n" );
document.write( "F1=F2=1,  Fn=F(n-1)+F(n-2) if n>2.<BR>\n" );
document.write( "Show that it holds for n which is greator and equal to 1.<BR>\n" );
document.write( "Fn < 2^(n)<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #14517 by </B><A HREF=/tutors/aboutme.mpl?userid=venugopalramana><B>venugopalramana(3286)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=venugopalramana><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.geocities.com/venugopalramana/FREE_MATHS_ASSISTANCE.html valign=middle><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=14517\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> LET US COMPARE THE GIVEN SERIES WITH A GEOMETRIC SERIES WHOSE WE CAN DO...<BR>\n" );
document.write( "THE G.P IS ...1,2,4,8,16,.......WITH C.R. OF 2 AND A=1.<BR>\n" );
document.write( "FIRSTLY WE FIND THAT THE GIVEN SERIES IS ALL POSITIVE NUMBERS AND INCREASING SERIES AFTER F2,SINCE EVERY TERM IS THE SUM OF PREVIOUS 2 TERMS WHICH ARE POSITIVE...JENCE..F2<F3<F4...ETC....<BR>\n" );
document.write( "NOW WE HAVE..<BR>\n" );
document.write( "F1=1............................................G1=1.........F1=G1<BR>\n" );
document.write( "F2=1............................................G2=2G1.......F2<G2<BR>\n" );
document.write( "F3=F1+F2<F2+F2=2F2..............................G3=2G2.......F3<G3<BR>\n" );
document.write( "F4=F2+F3<F3+F3=2F3..............................G4=2G3.......F4<G4<BR>\n" );
document.write( "....................................................................<BR>\n" );
document.write( ".....................................................................<BR>\n" );
document.write( "FN=F(N-2)+F(N-1)<F(N-1)+F(N-1)=2F(N-1)..........GN=2G(N-1)...FN<GN<BR>\n" );
document.write( "------------------------------------------------------------------------<BR>\n" );
document.write( "ADDING ALL THE ABOVE WE GET...(F1+F2+F3+F4+...+FN)<(G1+G2+G3+.....+GN)<BR>\n" );
document.write( "BUT WE KNOW THAT <BR>\n" );
document.write( "G1+G2+G3+.....+GN=1+2+4+8+.........2^(N-1)<BR>\n" );
document.write( "A=1...R=2...SO...SUM IS ..A*(R^N-1)/(R-1)=1*(2^N-1)/(2-1)=2^N-1<BR>\n" );
document.write( "HENCE <BR>\n" );
document.write( "(F1+F2+F3+F4+...+FN)<(G1+G2+G3+.....+GN)=2^N-1<2^N...THEN OBVIOUSLY <BR>\n" );
document.write( "FN<2^N\r<BR>\n" );
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document.write( "</TD></TR></TABLE>\n" );