For each set x, find the value of n(x). X is the set of positive, even natural numbers less than 75

To find \( n(X) \), where \( X \) is the set of positive, even natural numbers less than 75, we first list the numbers in the set \( X \).

### Set \( X \):
The even natural numbers less than 75 are:
\[
X = \{2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74\}
\]

### Step-by-step process:
1. The smallest even number is 2.
2. The largest even number less than 75 is 74.
3. These numbers form an arithmetic sequence with the first term \( a = 2 \), the common difference \( d = 2 \), and the last term \( l = 74 \).

The formula for the \( n \)-th term of an arithmetic sequence is:
\[
a_n = a + (n-1)d
\]
We can solve for \( n \) by plugging in the values for \( a = 2 \), \( d = 2 \), and \( a_n = 74 \):
\[
74 = 2 + (n-1) \cdot 2
\]
\[
74 = 2 + 2n - 2
\]
\[
74 = 2n
\]
\[
n = \frac{74}{2} = 37
\]

### Conclusion:
The number of elements in the set \( X \), denoted by \( n(X) \), is \( 37 \).

Do you want further details on how to apply the arithmetic sequence formula?

Here are 5 related questions:
1. How would you calculate \( n(X) \) if \( X \) included odd numbers?
2. How does the arithmetic sequence formula change with different common differences?
3. Can you apply the same method to find the number of odd numbers less than 75?
4. What if the set were even numbers greater than 75? How would that affect \( n(X) \)?
5. How many even numbers less than 100 are there?

**Tip:** Arithmetic sequences are great for quickly identifying the number of terms in regularly spaced sets, such as even or odd numbers.

For each set x, find the value of n(x). X is the set of positive, even natural numbers less than 75.

Learn how to find the number of positive, even natural numbers less than 75 using arithmetic sequences. This solution walks you through the process step by step, covering the formula for the n-th term and applying it to the set of even numbers. Suitable for Grades 6-8 students.

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