tmua.me Extreme Set B Paper 1 · Question 10

Initially a list has the single entry $\dfrac12$ . In each round, every entry $x$ is replaced simultaneously by the two entries

\frac{x}{3}\qquad\text{and}\qquad\frac{x+2}{3}.

After eight rounds the list has $256$ entries. For each entry $x$ , the value $x(1-x)$ is calculated.

What is the sum of these $256$ values?

16\bigl(1+3^{-16}\bigr)

32\bigl(1+3^{-16}\bigr)

32\bigl(1-3^{-16}\bigr)

32\bigl(1+3^{-8}\bigr)

32\bigl(1-3^{-8}\bigr)

16\bigl(1+3^{-8}\bigr)

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