The constant a is such that \(\int_0^a xe^{-2x} \, dx = \frac{1}{8}\).

(a) Show that \(a = \frac{1}{2} \ln(4a + 2)\).

(b) Verify by calculation that a lies between 0.5 and 1.

(c) Use an iterative formula based on the equation in (a) to determine a correct to 2 decimal places. Give the result of each iteration to 4 decimal places.