Start by plugging in 0.1 for x in the equation
for f(x). Then take what you get, and plug that
in for x in the equation for f(x).
Keep taking what you get each time and plugging
that in for x in the equation for f(x) over and over.
f(x)=1.2x[1-x], x0 = 0.1
f(x0) = f(0.1)=1.2(0.1)[1-(0.1)] = 0.108 = x1
f(x1) = f(0.108)=1.2(0.108)[1-(0.108)] = 0.1156032 = x2
f(x2) = f(0.1156032)=1.2(0.1156032)[1-(0.1156032)] = 0.1226869202 = x3
f(x3) = f(0.1226869202)=1.2(0.1226869202)[1-(0.1226869202)] = 0.1291618079 = x4
f(x4) = f(0.1291618079)=1.2(0.1291618079)[1-(0.1291618079)] = 0.1349748422 = x5
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Interesting fact FYI:
If you keep getting iterates for this, they will get closer
and closer to 0.1666666... which gets closer and closer to the
fraction 1/6.
You can tell this by setting f(x) = x and solving
1.2x[1-x] = x
1.2x-1.2x² = x
-1.2x²+1.2x-x = 0
1.2x²-1.2x+x = 0
12x²-12x+10x = 0
12x²-2x = 0
6x²-x = 0
x(6x-1) = 0
x=0; 6x-1 = 0
6x = 1
x = 1/6
If we were beginning with x0 = 0 we would of
course get 0, but if we started with any number x0
between 0 and 1, 0 < x0 < 1 the iterations would always
converge to (get closer and closer to) 1/6.
Edwin
