Let X be the set consisting of the first 2018 terms of the arithmetic progression 1,6, 11,.... and Y be the set consisting of the first 2018 terms of the arithmetic progression 9,16,23,... Then, the number of elements in the set `X uu U ` is ____

Answer

To solve the problem, we need to find the number of distinct elements in the union of two sets \( X \) and \( Y \), where:

- \( X \) consists of the first 2018 terms of the arithmetic progression (AP) \( 1, 6, 11, \ldots \)
- \( Y \) consists of the first 2018 terms of the arithmetic progression \( 9, 16, 23, \ldots \)

### Step 1: Determine the terms in set \( X \)

The first term of the AP \( X \) is \( a_1 = 1 \) and the common difference \( d = 5 \).

The \( n \)-th term of an AP can be calculated using the formula:
\[ a_n = a_1 + (n - 1) \cdot d \]

For \( n = 2018 \):
\[ a_{2018} = 1 + (2018 - 1) \cdot 5 = 1 + 2017 \cdot 5 = 1 + 10085 = 10086 \]

Thus, the last term in set \( X \) is \( 10086 \).

### Step 2: Determine the terms in set \( Y \)

The first term of the AP \( Y \) is \( b_1 = 9 \) and the common difference \( d = 7 \).

Using the same formula for the \( n \)-th term:
\[ b_n = b_1 + (n - 1) \cdot d \]

For \( n = 2018 \):
\[ b_{2018} = 9 + (2018 - 1) \cdot 7 = 9 + 2017 \cdot 7 = 9 + 14119 = 14128 \]

Thus, the last term in set \( Y \) is \( 14128 \).

### Step 3: Find the common elements in sets \( X \) and \( Y \)

To find the common elements, we need to determine the values that are present in both sequences.

The general term for set \( X \) is:
\[ x_n = 1 + (n - 1) \cdot 5 = 5n - 4 \]

The general term for set \( Y \) is:
\[ y_m = 9 + (m - 1) \cdot 7 = 7m + 2 \]

We need to find integers \( n \) and \( m \) such that:
\[ 5n - 4 = 7m + 2 \]
Rearranging gives:
\[ 5n - 7m = 6 \]

### Step 4: Solve the Diophantine equation

This is a linear Diophantine equation. We can find integer solutions for \( n \) and \( m \).

Using the method of finding particular solutions, we can try different values of \( n \) and \( m \) to find solutions.

After solving, we find that the general solution can be expressed as:
\[ n = 6 + 7k \quad \text{and} \quad m = 4 + 5k \quad \text{for integer } k \]

### Step 5: Determine the range of \( k \)

To find the valid values of \( k \), we need \( n \) to be between \( 1 \) and \( 2018 \):
\[ 1 \leq 6 + 7k \leq 2018 \]
This simplifies to:
\[ -5 \leq 7k \leq 2012 \implies 0 \leq k \leq 287 \]

Thus, \( k \) can take values from \( 0 \) to \( 287 \), giving \( 288 \) common terms.

### Step 6: Calculate the number of distinct elements in \( X \cup Y \)

Using the principle of inclusion-exclusion:
\[ |X \cup Y| = |X| + |Y| - |X \cap Y| \]
Where:
- \( |X| = 2018 \)
- \( |Y| = 2018 \)
- \( |X \cap Y| = 288 \)

Thus:
\[ |X \cup Y| = 2018 + 2018 - 288 = 3748 \]

### Final Answer

The number of elements in the set \( X \cup Y \) is **3748**.