Multiple choice

If the mean of '$n$' observations, $1^2, 2^2, 3^2, ... , n^2$ is $\dfrac{46 n}{11}$, find the value of '$n$'.

  1. $22$
  2. $23$
  3. $12$
  4. $11$
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D Correct answer
Explanation

Sum of squares = n(n+1)(2n+1)/6. Mean = (n+1)(2n+1)/6 = 46n/11. This leads to a cubic equation. Testing n=11: (12*23)/6 = 2*23 = 46. 46*11/11 = 46. Matches.

AI explanation

Using the formula for the sum of squares of the first n natural numbers, the total sum is n(n plus 1)(2n plus 1) divided by 6. Dividing by n gives a mean of (n plus 1)(2n plus 1) divided by 6, which is set equal to 46n divided by 11. Cross multiplying yields 11 times (2n squared plus 3n plus 1) equals 276n, and simplifying gives 22n squared minus 243n plus 11 equals 0; solving this quadratic equation gives n equal to 11.

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