(i)[5]
If $\int_0^a (3e^{\frac{1}{2}x} + 1)\,dx = 10$, demonstrate that the positive constant $a$ obeys $a = 2 \ln\left(\frac{16 - a}{6}\right)$.
(ii)[3]
Apply the iterative formula $a_{n+1} = 2 \ln\left(\frac{16 - a_n}{6}\right)$ with $a_1 = 2$ to determine $a$ correct to $3$ decimal places. Record every iteration result to $5$ decimal places.