Question

The average of the squares of the first 50 natural numbers is

• A. \( \dfrac{1817}{2} \)
• B. \( \dfrac{1717}{2} \)
• C. \( \dfrac{1517}{2} \)
• D. \( \dfrac{1617}{2} \)

Hint:

The sum of squares of the first \( n \) natural numbers can be calculated using the formula \( \dfrac{n(n + 1)(2n + 1)}{6} \), and the average is obtained by dividing the sum by \( n \).

Answer 1

The Correct Option is B

Step 1: Formula for the average of squares.
The average of the squares of the first \( n \) natural numbers is given by the formula: \[ \text{Average} = \dfrac{1^2 + 2^2 + 3^2 + \dots + n^2}{n}. \] The sum of the squares of the first \( n \) natural numbers is given by: \[ \text{Sum of squares} = \dfrac{n(n + 1)(2n + 1)}{6}. \]

Step 2: Apply the formula for \( n = 50 \).
Substitute \( n = 50 \) into the formula: \[ \text{Sum of squares} = \dfrac{50(50 + 1)(2 \times 50 + 1)}{6} = \dfrac{50 \times 51 \times 101}{6} = 42925. \] Now, the average is: \[ \text{Average} = \dfrac{42925}{50} = \dfrac{1717}{2}. \]

Step 3: Conclusion.
Thus, the average of the squares of the first 50 natural numbers is \( \dfrac{1717}{2} \).

Final Answer: \( \dfrac{1717}{2} \).