Deriving the expected number of steps that is taken to perform the k'th operation
Let $N_i$ be the random variable denoting the length of $T$ prior to the $i$th operation. You should be able to calculate $\mathbb{E}[N_i]$.

The expected running time $T_i$ of the $i$th operation is
$$
p \cdot 1 + (1-p) \cdot \frac{1+\cdots+N_i}{N_i} = p + (1-p) \frac{N_i+1}{2}.
$$
Given $\mathbb{E}[N_i]$, you should be able to calculate $\mathbb{E}[T_i]$ using this formula.

I leave you to figure out all remaining details.