<s>[INST] Something maybe related to stochastic process We have $5$ floors and we are on the ground(treat the ground as floor$0$).
The probability of jumping from the ground to floor$1$ is $1$.
For $i\in\{1,2,3,4\}$,
the probability of jumping from the $i$-th floor to the $i+1$-th floor is $\frac{1}{3^i}$,
the probability of staying at the $i$-th floor is $\frac{3^i-1}{2\times 3^i}$,
the probability of jumping from the $i$-th floor to the $i-1$-th floor is $\frac{3^i-1}{2\times 3^i}$,
What is the expectation of the number of turns that you jump from the ground and reach floor$5$?
My thought:
I first came up with the probability matrix $P = \begin{pmatrix}
 0&  \frac{1}{3}&  0&  0&  0& 0\\ 
 1&  \frac{1}{3}&  \frac{4}{9}&  0&  0& 0\\ 
 0&  \frac{1}{3}&  \frac{4}{9}&  \frac{13}{27}&  0& 0\\ 
 0&  0&  \frac{1}{9}&  \frac{13}{27}&  \frac{40}{81}& 0\\ 
 0&  0&  0&  \frac{1}{27}&  \frac{40}{81}& 0\\ 
 0&  0&  0&  0&  \frac{1}{81}& 0
\end{pmatrix}$
and the initial state is $e_1 = \begin{pmatrix}
1\\ 
0\\ 
0\\ 
0\\ 
0\\ 
0
\end{pmatrix}$
so the expectation should be
$$\sum_{x=0}^\infty x[(P^xe_1)\cdot e_6]$$ where $\cdot$ is the inner product and $e_6 = \begin{pmatrix}
0\\ 
0\\ 
0\\ 
0\\ 
0\\ 
1
\end{pmatrix}$
Is my thought correct so far? And I can't go through it further. I used programming and got an approximation of the series that the expectation should be $14543$
 [/INST] Labelling floors $0 - 5$ as $a - f$, and using first-step-analysis, we have
$\displaylines{a = 1+b\\  
b= 1+c/3 + (1/3)(b+a)\\  
c= 1+d/9 + (4/9)(c+b)\\  
d=1+e/27 +(13/27)(d+c)\\  
e = 1 +(40/81)(e+d)}$
and wolfram confirms your answer of $\;\fbox{14543}$
</s>