To solve the problem, we need to find the median of the sequence \( x_1, x_2, x_3, \ldots, x_{34} \) defined by the given conditions.
### Step 1: Identify the first 9 terms
From the problem, we know that:
- For \( i = 1, 2, \ldots, 9 \), \( x_i = 150 \).
Thus, the first 9 terms are:
\[
x_1 = x_2 = x_3 = \ldots = x_9 = 150
\]
### Step 2: Determine the next terms from \( x_{10} \) to \( x_{34} \)
From \( i = 10 \) to \( i = 33 \), we have the relationship:
\[
x_{i+1} - x_i = -2 \implies x_{i+1} = x_i - 2
\]
Starting from \( x_{10} \):
\[
x_{10} = 150 \quad \text{(since \( x_9 = 150 \))}
\]
Now we can calculate the subsequent terms:
- \( x_{11} = x_{10} - 2 = 150 - 2 = 148 \)
- \( x_{12} = x_{11} - 2 = 148 - 2 = 146 \)
- \( x_{13} = x_{12} - 2 = 146 - 2 = 144 \)
- \( x_{14} = x_{13} - 2 = 144 - 2 = 142 \)
- \( x_{15} = x_{14} - 2 = 142 - 2 = 140 \)
- \( x_{16} = x_{15} - 2 = 140 - 2 = 138 \)
- \( x_{17} = x_{16} - 2 = 138 - 2 = 136 \)
- \( x_{18} = x_{17} - 2 = 136 - 2 = 134 \)
- \( x_{19} = x_{18} - 2 = 134 - 2 = 132 \)
- \( x_{20} = x_{19} - 2 = 132 - 2 = 130 \)
- \( x_{21} = x_{20} - 2 = 130 - 2 = 128 \)
- \( x_{22} = x_{21} - 2 = 128 - 2 = 126 \)
- \( x_{23} = x_{22} - 2 = 126 - 2 = 124 \)
- \( x_{24} = x_{23} - 2 = 124 - 2 = 122 \)
- \( x_{25} = x_{24} - 2 = 122 - 2 = 120 \)
- \( x_{26} = x_{25} - 2 = 120 - 2 = 118 \)
- \( x_{27} = x_{26} - 2 = 118 - 2 = 116 \)
- \( x_{28} = x_{27} - 2 = 116 - 2 = 114 \)
- \( x_{29} = x_{28} - 2 = 114 - 2 = 112 \)
- \( x_{30} = x_{29} - 2 = 112 - 2 = 110 \)
- \( x_{31} = x_{30} - 2 = 110 - 2 = 108 \)
- \( x_{32} = x_{31} - 2 = 108 - 2 = 106 \)
- \( x_{33} = x_{32} - 2 = 106 - 2 = 104 \)
### Step 3: Determine \( x_{34} \)
Since \( x_{34} \) is not defined by the previous relation, we can assume it continues the pattern:
\[
x_{34} = x_{33} - 2 = 104 - 2 = 102
\]
### Step 4: List all terms
Now we have the complete list of terms:
\[
x_1 = x_2 = \ldots = x_9 = 150, \quad x_{10} = 150, \quad x_{11} = 148, \quad x_{12} = 146, \ldots, \quad x_{33} = 104, \quad x_{34} = 102
\]
### Step 5: Find the median
To find the median of 34 terms, we need to calculate the average of the 17th and 18th terms:
- The first 10 terms are \( 150 \).
- The next terms decrease by 2 starting from \( 150 \).
The 17th term is:
\[
x_{17} = 136
\]
The 18th term is:
\[
x_{18} = 134
\]
Calculating the median:
\[
\text{Median} = \frac{x_{17} + x_{18}}{2} = \frac{136 + 134}{2} = \frac{270}{2} = 135
\]
### Final Answer
The median of \( x_1, x_2, \ldots, x_{34} \) is \( \boxed{135} \).
