The set $\{1, 4, 9, 16, 25, \ldots\}$ in set-builder form is:
A$\{x : x \text{ is a multiple of } 4\}$
B$\{x : x = n^{2}, \text{ where } n \in \mathbb{N}\}$
C$\{x : x = 2n, \text{ where } n \in \mathbb{N}\}$
D$\{x : x \text{ is odd and prime}\}$
Answer & Solution
Correct answer: B. $\{x : x = n^{2}, \text{ where } n \in \mathbb{N}\}$
The pattern $1, 4, 9, 16, 25, \ldots$ is $1^{2}, 2^{2}, 3^{2}, 4^{2}, 5^{2}, \ldots$ — i.e. the **squares of natural numbers**.
In set-builder form: $\{x : x = n^{2}, n \in \mathbb{N}\}$.
Equivalently: $\{x : x \text{ is the square of a natural number}\}$.
- Trap A gives multiples of $4$: $\{4, 8, 12, 16, 20, \ldots\}$ — different set.
- Trap C gives evens: $\{2, 4, 6, 8, \ldots\}$.
- Trap D gives $\{3, 5, 7, 11, \ldots\}$.
NCERT Class 11, Example 3.
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