I want to compute and understand $E[T_{n+1}|T_n]$ where $T_n$ is the number of flips on a fair coin until I get $n$ consecutive heads.
I think I know how to compute this. If we denote by $X_k$ the flip of the $k$-coin we have:
\begin{align*}\displaystyle T_{n+1}=&T_{n+1} \cdot \boldsymbol{1}_{\{X_{T_{n}+1}=1\}}+T_{n+1}\boldsymbol{1}_{\{X_{T_{n}+1}=0\}}\\ \displaystyle =&(T_{n}+1) \cdot \boldsymbol{1}_{\{X_{T_{n}+1}=1\}}+(T_{n}+1+T_{n+1})\boldsymbol{1}_{\{X_{T_{n}+1}=0\}} \end{align*} where $\boldsymbol{1}_A$ is the indicator function of the event $A$. Taking conditional expectations $E[\,\cdot\,| T_n]$ at both sides and using the independence of the different flips we get
\begin{align*}\displaystyle E[T_{n+1}|T_n]=&\frac{1}{2}E[T_{n}+1|T_n]+\frac{1}{2}E[T_{n}+1+T_{n+1}|T_n]\\ \displaystyle =&\frac{1}{2}(T_{n}+1)+\frac{1}{2}\big(T_{n}+1+E[T_{n+1}|T_n]\big) \end{align*} which gives $E[T_{n+1}|T_n]=2(T_n+1)$.
But I don't know how to understand this equation. Let's take $n=1$, then we are trying to understand $E[T_{2}|T_1]=2(T_1+1)$.
Now, let's say that I am running an experiment and I had to flip 100 times my coin until getting one head (extremely unlucky). Then, $T_1=100$. Once I got my first head, I want to guess now the total number of coins until I get two consecutive heads. As $T_1=100$, does this mean that the expected number of flips for two consecutive heads is $2(100+1)=202$? That seems wrong.
Maybe I'm misunderstanding the concept of the conditional expectation, or something is wrong in the computation, as something like $E[T_{2}|T_1]=T_1+E[T_1]+2$ would make more sense to me.