1) 1/9
2) let x=0.413413413...
1000x=413.413413...
x=0.413413413...
1000x-x=999x
999x=413
x=413/999
3) let x=0.2161616...
1000x=216.161616...
10x=2.161616...
1000x-10x=990x
990x=214
x=214/990
x=107/495
I just felt like answering them :)
5) let roots be t and t+1
SUM OF ROOTS:
t+t+1=3
2t=2
t=1
therefore roots are 1 and 2
x^2-3x+(k+1)=0
(x-2)(x-1)=0
(k+1)=-1 x -2
k+1=2
k=1
(x-2)^2 + (y-5)^2 = (y+3)^2
x^2 -4x+4 +y^2-10y+25 = y^2 +6y +9
x^2 -4x -16y+20=0