Question

The set defined by \(A = \{ n \in \mathbb{N} \mid (1 + n^2) < 50 \}\) , where N is the set of natural numbers, then the mean value of elements of A is

• A. 1
• B. 6
• C. 4
• D. 3.5

Answer 1

The Correct Option is D

We are given the set \( A = \{n \in \mathbb{N} \mid \frac{1}{1 + n^2} < 50 \} \), which means that for each element \( n \) of \( A \), we have the condition: \[ \frac{1}{1 + n^2} < 50 \] First, we solve for \( n \). Rearranging the inequality: \[ 1 + n^2 > \frac{1}{50} \] \[ n^2 > \frac{1}{50} - 1 = \frac{-49}{50} \] Since \( n^2 \) is always positive for natural numbers, the inequality holds for all natural numbers. Next, we observe that the elements of \( A \) must be natural numbers. We list the elements of \( A \) that satisfy the condition \( 1 + n^2 < 50 \), i.e., \( n^2 < 49 \). The values of \( n \) satisfying \( n^2 < 49 \) are: \[ n = 1, 2, 3, 4, 5, 6 \] Thus, the elements of the set \( A \) are \( \{1, 2, 3, 4, 5, 6\} \). To find the mean value of the elements of \( A \), we calculate the average: \[ \text{Mean} = \frac{1 + 2 + 3 + 4 + 5 + 6}{6} = \frac{21}{6} = 3.5 \] Thus, the mean value of the elements of \( A \) is \( 3.5 \).

The correct option is (D): \(3.5\)