A more game related stats question There are an infinite amount of true-false questions. My goal is to get to 100 points. 
A correct answer will give me 1 point, and if I have 2 questions correct in a row, the any correct questions that are proceded by  2 correct questions will give me an additional point. So it's like a winning streak. 
(Example: right, wrong, wrong, right, right, right, wrong. I receive 5 points)
A incorrect answer will have no penalty except ending the bonus streak. 
The question is, if I completely guess all the questions, so 50%, how many questions do I expect to play in order to get to 100 points? 
Thanks all! 
 A: Let's call $a_n$, $b_n$ and $c_n$ the expected number of questions to play before getting $n$ points if you previously got $0$, $1$ or at least $2$ answers right, respectively. So the value you're looking for is $a_{100}$. Then we have the recurrence relations
\begin{align}
a_n&=1+\frac12(a_n+b_{n-1})\;,\\
b_n&=1+\frac12(a_n+c_{n-1})\;,\\
c_n&=1+\frac12(a_n+c_{n-2})\;.
\end{align}
Solving the first equation for $b_{n-1}$ yields $b_{n-1}=a_n-2$, which we can substitute into the second equation to obtain $a_{n+1}-2=1+\frac12(a_n+c_{n-1})$. Solving for $c_{n-1}$ yields $c_{n-1}=2a_{n+1}-a_n-6$, and then substituting into the third equation yields
$$
2a_{n+2}-a_{n+1}-6=1+\frac12(a_n+2a_n-a_{n-1}-6)\;,
$$
or
$$
a_n-\frac12a_{n-1}-\frac34a_{n-2}+\frac14a_{n-3}-2=0\;.
$$
The ansatz $a_n=\mu n$ yields $\mu=\frac85$, and the characteristic equation $\lambda^3-\frac12\lambda^2-\frac34\lambda+\frac14=0$ has roots $1$ and $\lambda_\pm=(-1\pm\sqrt5)/4$ (with $|\lambda_\pm|\lt1$), so the solution has the general form
$$
a_n=\frac85n+c+c_+\left(\frac{-1+\sqrt5}4\right)^n+c_-\left(\frac{-1-\sqrt5}4\right)^n\;.
$$
Thus we have $a_{100}\approx160$, and you can get the exact value by finding $c$ and $c_\pm$ using the initial values $a_0=b_0=c_0=c_{-1}=0$.
