Question

Let \(X\) be the set consisting of the first 2018 terms of the arithmetic progression \(1,6,11, \ldots\), and \(Y\) be the set consisting of the first 2018 terms of the arithmetic progression \(9,16,23, \ldots\). Then, the number of elements in the set \(X \cup Y\) is

Short answer

The number of elements in the set \(X \cup Y\) is 3633.

Step-by-step solution

Identifying the Formula for the nth Term

For an arithmetic progression (AP), the nth term is given by the formula: \(a_n = a_1 + (n-1)d\), where \(a_1\) is the first term and \(d\) is the common difference. For the set \(X\), the first term is 1 and the common difference is 5. For the set \(Y\), the first term is 9 and the common difference is 7.

Finding the Last Term of Each Sequence

For set \(X\), the 2018th term is \(1 + (2017 \times 5) = 10086\), and for set \(Y\), it is \(9 + (2017 \times 7) = 14120\). These are the last terms of each AP.

Finding the Intersection of X and Y

Determine if there are common terms in both sequences. The common terms will follow the condition \(1 + 5k = 9 + 7l\) for some integers \(k\) and \(l\). Rearranging gives us \(2l = 5k + 4\). Among the integers between 1 and 2018, we need to find the number of solutions to this equation.

Counting the Number of Common Elements

Since \(l\) must be even (because 2l is an integer), let \(l = 2m\). This gives us the equation \(5k = 4m + 4\), or \(k = \frac{4m + 4}{5}\). We need to count the number of even numbers \(m\) such that \(k\) is also an integer. By iterating over the possible values, we see that there is a common term for every 5 terms from the second sequence. So the number of common terms will be \(\lfloor \frac{2018}{5} \rfloor\), where \(\lfloor \cdot \rfloor\) denotes the floor function.

Calculating the Number of Elements in the Union

The number of elements in the union of \(X\) and \(Y\) is the sum of the elements in each set minus the number of common elements: \(\#(X \cup Y) = \#X + \#Y - \#(X \cap Y) = 2018 + 2018 - \lfloor \frac{2018}{5} \rfloor\).

Final Calculation

Perform the calculation to get the final answer: \(\#(X \cup Y) = 2018 + 2018 - \lfloor \frac{2018}{5} \rfloor = 4036 - 403 = 3633\).

Key concepts

Nth Term of an Arithmetic Progression (AP)

Understanding the nth term of an arithmetic progression is key to grasping the entire concept of these sequences. The general formula for the nth term is given by
\( a_n = a_1 + (n-1)d \)
where \(a_1\) is the first term, \(d\) is the common difference, and \(n\) refers to the nth term in the progression.

Let's look at an example to illustrate this. Suppose you want to find the 50th term in an AP where the first term is 3 and the common difference is 2. Using the formula, you would calculate it as
\( a_{50} = 3 + (50-1) \times 2 = 3 + 49 \times 2 = 101 \).

Remember, the nth term helps you pinpoint any term in the sequence without listing all the preceding terms, which is especially useful for long sequences.

Set Theory in Mathematics

Set theory is a fundamental part of mathematics that deals with the collection of objects, known as 'sets'. In our case, understanding sets helps us handle large quantities of numbers efficiently and perform operations like union
(\(X \bigcup Y\)) and intersection
(\(X \bigcap Y\)).

In terms of sequences, each AP can be thought of as a set of numbers. For instance, if an AP is described as \(1, 6, 11, \text{...}\), we could define a set \(X\) that contains the first 2018 terms of this sequence. Similarly, another AP, say \(9, 16, 23, \text{...}\), forms another set \(Y\).

With set theory, we use the concepts of unions and intersections to find how many unique elements exist between two sets or how many elements they share, respectively. This is crucial for analyzing overlapping sequences and finding the total count of distinct elements in combined sets as seen in the example problem.

Arithmetic Sequences

An arithmetic sequence is essentially a list of numbers with a specific pattern; each term is created by adding a fixed number to the previous term. This fixed number is called the 'common difference'.

For example, in the sequence \(2, 5, 8, 11, \text{...}\), the common difference is 3. Arithmetic sequences are everywhere in daily life, such as in the scoring system of a game or the pattern in which leaves are arranged on a stem.

Arithmetic sequences can be finite or infinite depending on whether they have a fixed number of terms or continue indefinitely. Understanding arithmetic sequences allows us to predict future terms, sum a series of terms, and even solve complex problems related to time, speed, and distance.

In fact, this concept is so versatile that it can be extended to science, finance, and other areas where patterns and rates of change are studied.