You know that f(x)=1-2x
f(f(x)) = f( 1- 2x) = 1 - 2(1-2x) = 1 - 2 + 4x = -1 + 4x
f(f(1))+..+f(f(5))
=> -1 + 4*1 - 1 + 4*2...-1 + 4*5
=> -1*5 + 4( 1+ 2+3+4+5)
=> -5 + 4*15
=> -5 + 60
=> 55
The required result of f(f(1))+..+f(f(5)) = 55
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