An Iterated Fraction

Let $f(x) = \dfrac{x + a}{x + 1}$ for $x > 0$, and suppose the tangent line to $y = f(x)$ at $(1, f(1))$ has $y$-intercept $\dfrac{11}{2}$.

1. Find $a$. 2. Discuss the monotonicity of $g(x) = x \cdot f^2(x)$. 3. Let $a_1 = 1$ and $a_{n+1} = f(a_n)$ for $n \in \mathbb{N}^*$. Prove that