Question 26646
LET US COMPARE THE GIVEN SERIES WITH A GEOMETRIC SERIES WHOSE WE CAN DO...
THE G.P IS ...1,2,4,8,16,.......WITH C.R. OF 2 AND A=1.
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....
NOW WE HAVE..
F1=1............................................G1=1.........F1=G1
F2=1............................................G2=2G1.......F2<G2
F3=F1+F2<F2+F2=2F2..............................G3=2G2.......F3<G3
F4=F2+F3<F3+F3=2F3..............................G4=2G3.......F4<G4
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FN=F(N-2)+F(N-1)<F(N-1)+F(N-1)=2F(N-1)..........GN=2G(N-1)...FN<GN
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ADDING ALL THE ABOVE WE GET...(F1+F2+F3+F4+...+FN)<(G1+G2+G3+.....+GN)
BUT WE KNOW THAT 
G1+G2+G3+.....+GN=1+2+4+8+.........2^(N-1)
A=1...R=2...SO...SUM IS ..A*(R^N-1)/(R-1)=1*(2^N-1)/(2-1)=2^N-1
HENCE 
(F1+F2+F3+F4+...+FN)<(G1+G2+G3+.....+GN)=2^N-1<2^N...THEN OBVIOUSLY 
FN<2^N