Write the set X=`{1,1/4,1/9,(1)/(16),(1)/(25),…}` in the set-builder form.

Answer

To express the set \( X = \{ 1, \frac{1}{4}, \frac{1}{9}, \frac{1}{16}, \frac{1}{25}, \ldots \} \) in set-builder notation, we can follow these steps:

### Step 1: Identify the elements of the set
The elements of the set are:
- \( 1 \)
- \( \frac{1}{4} \) (which is \( \frac{1}{2^2} \))
- \( \frac{1}{9} \) (which is \( \frac{1}{3^2} \))
- \( \frac{1}{16} \) (which is \( \frac{1}{4^2} \))
- \( \frac{1}{25} \) (which is \( \frac{1}{5^2} \))
- and so on.

### Step 2: Recognize the pattern
We can see that the denominators of the fractions are perfect squares of natural numbers:
- The first term is \( 1 = \frac{1}{1^2} \)
- The second term is \( \frac{1}{2^2} \)
- The third term is \( \frac{1}{3^2} \)
- The fourth term is \( \frac{1}{4^2} \)
- The fifth term is \( \frac{1}{5^2} \)
- This pattern continues indefinitely.

### Step 3: Define the variable
Let \( n \) be a natural number. The general term for the elements of the set can be expressed as:
\[ Y = \frac{1}{n^2} \]

### Step 4: Write the set in set-builder notation
Now we can express the set \( X \) in set-builder notation. We can say:
\[ X = \{ Y \mid Y = \frac{1}{n^2}, n \in \mathbb{N} \} \]

### Final Answer
Thus, the set \( X \) in set-builder form is:
\[ X = \{ Y \mid Y = \frac{1}{n^2}, n \in \mathbb{N} \} \]

---