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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