Derive expectation for two consecutive heads

Let $T$ denote the number of flips of a fair coin needed until the pattern HH (two consecutive heads) occurs for the first time. (a) Compute $E[T]$ using a state-based recursion or a Markov-chain argument, and give the exact numerical value. (b) Generalize $E[T]$ to the case of a biased coin with $P(H)=p$, where $0<p<1$. (c) For the fair coin, derive $\mathrm{Var}(T)$. Show all steps and justify the recurrence setup.

Overview: This question assesses command of probability theory and stochastic processes, in particular expectation and variance calculations, Markov-chain or recurrence reasoning, and pattern waiting times in Bernoulli trials, as relevant for a Data Scientist role.