Union of Two Arithmetic Progressions Term Sets

To find the number of distinct elements in the set $X \cup Y$, we can use the Principle of Inclusion-Exclusion: $n(X \cup Y) = n(X) + n(Y) - n(X \cap Y)$

Step 1: Analyze set $X$

Set $X$ consists of the first $2018$ terms of the arithmetic progression with:

• First term, $a_1 = 1$
• Common difference, $d_1 = 6 - 1 = 5$
• Total number of elements, $n(X) = 2018$

The general term of this AP is given by: $x_k = 1 + (k - 1) \cdot 5 = 5k - 4, \quad \text{for } k = 1, 2, \dots, 2018$

The maximum element in set $X$ is: $x_{\max} = 1 + (2018 - 1) \cdot 5 = 1 + 2017 \cdot 5 = 10086$

Step 2: Analyze set $Y$

Set $Y$ consists of the first $2018$ terms of the arithmetic progression with:

• First term, $a_2 = 9$
• Common difference, $d_2 = 16 - 9 = 7$
• Total number of elements, $n(Y) = 2018$

The general term of this AP is given by: $y_m = 9 + (m - 1) \cdot 7 = 7m + 2, \quad \text{for } m = 1, 2, \dots, 2018$

The maximum element in set $Y$ is: $y_{\max} = 9 + (2018 - 1) \cdot 7 = 9 + 2017 \cdot 7 = 14128$

Step 3: Find the number of common elements, $n(X \cap Y)$

The elements in $X \cap Y$ form a new arithmetic progression:

• The first common term is $16$ (since $x_4 = 5(4) - 4 = 16$ and $y_2 = 7(2) + 2 = 16$).
• The common difference of the common terms is $D = \operatorname{lcm}(d_1, d_2) = \operatorname{lcm}(5, 7) = 35$.

Thus, the common elements are of the form: $z_t = 16 + 35t, \quad \text{for } t = 0, 1, 2, \dots$

Since the maximum possible value for a common element cannot exceed the smaller maximum of the two sets: $\max(X \cap Y) \le \min(x_{\max}, y_{\max}) = 10086$

Setting up the inequality: $16 + 35t \le 10086$ $35t \le 10070$ $t \le \frac{10070}{35} = 287.714\dots$

Since $t$ must be a non-negative integer, $t$ can take values from $0$ to $287$.

The total number of common elements is: $n(X \cap Y) = 287 - 0 + 1 = 288$

Step 4: Calculate $n(X \cup Y)$

Using the Inclusion-Exclusion formula: $n(X \cup Y) = 2018 + 2018 - 288 = 3748$