To find the number of elements in the set \( X \cup Y \), we will follow these steps:
### Step 1: Identify the arithmetic progressions (APs)
The first AP, \( X \), consists of the first 2018 terms of the sequence:
\[ 1, 6, 11, 16, \ldots \]
The first term \( a_1 = 1 \) and the common difference \( d_1 = 5 \).
The \( n \)-th term of an AP can be calculated using the formula:
\[ a_n = a_1 + (n-1)d \]
Thus, the \( n \)-th term of \( X \) is:
\[ a_n = 1 + (n-1) \cdot 5 = 5n - 4 \]
The last term of \( X \) (when \( n = 2018 \)) is:
\[ a_{2018} = 5 \cdot 2018 - 4 = 10090 - 4 = 10086 \]
The second AP, \( Y \), consists of the first 2018 terms of the sequence:
\[ 9, 16, 23, \ldots \]
The first term \( b_1 = 9 \) and the common difference \( d_2 = 7 \).
Thus, the \( n \)-th term of \( Y \) is:
\[ b_n = 9 + (n-1) \cdot 7 = 7n + 2 \]
The last term of \( Y \) (when \( n = 2018 \)) is:
\[ b_{2018} = 7 \cdot 2018 + 2 = 14126 + 2 = 14128 \]
### Step 2: Find the common terms in \( X \) and \( Y \)
To find the common terms between the two APs, we need to set the general terms equal to each other:
\[ 5n - 4 = 7m + 2 \]
Rearranging gives:
\[ 5n - 7m = 6 \]
### Step 3: Solve for integer solutions
This is a linear Diophantine equation. We can find integer solutions for \( n \) and \( m \).
1. Rearranging gives:
\[ 5n = 7m + 6 \]
\[ n = \frac{7m + 6}{5} \]
2. For \( n \) to be an integer, \( 7m + 6 \) must be divisible by 5. We can find \( m \) values that satisfy this condition:
\[ 7m + 6 \equiv 0 \ (\text{mod} \ 5) \]
Simplifying gives:
\[ 2m + 1 \equiv 0 \ (\text{mod} \ 5) \]
\[ 2m \equiv 4 \ (\text{mod} \ 5) \]
\[ m \equiv 2 \ (\text{mod} \ 5) \]
Thus, \( m \) can be expressed as:
\[ m = 5k + 2 \] for integer \( k \).
### Step 4: Find the range of \( m \)
Since \( m \) must be a positive integer and \( m \) corresponds to the terms of \( Y \):
\[ 1 \leq m \leq 2018 \]
Substituting:
\[ 1 \leq 5k + 2 \leq 2018 \]
This simplifies to:
\[ -1 \leq 5k \leq 2016 \]
\[ 0 \leq k \leq 403.2 \]
Thus, \( k \) can take values from \( 0 \) to \( 403 \), giving \( 404 \) possible values for \( k \).
### Step 5: Calculate the number of common terms
For each \( k \), we can find the corresponding \( n \):
\[ n = \frac{7(5k + 2) + 6}{5} = 7k + 4 \]
This \( n \) must also satisfy \( 1 \leq n \leq 2018 \). The maximum value of \( k \) remains \( 403 \) since \( n \) will also be valid for all \( k \) in that range.
Thus, the number of common terms \( |X \cap Y| = 404 \).
### Step 6: Calculate the number of elements in \( X \cup Y \)
Using the principle of inclusion-exclusion:
\[ |X \cup Y| = |X| + |Y| - |X \cap Y| \]
Where:
- \( |X| = 2018 \)
- \( |Y| = 2018 \)
- \( |X \cap Y| = 404 \)
Thus:
\[ |X \cup Y| = 2018 + 2018 - 404 = 3632 \]
### Final Answer
The number of elements in the set \( X \cup Y \) is \( 3632 \).
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