Multiple choice

The mean of n observations 1, 4, 9, 16, ···, n2 is 130. What is the value of n?

  1. 18

  2. 19

  3. 20

  4. 21
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Sum of first n squares = n(n+1)(2n+1)/6. Mean = [n(n+1)(2n+1)/6] / n = (n+1)(2n+1)/6 = 130. (n+1)(2n+1) = 780. 2n^2 + 3n + 1 = 780. 2n^2 + 3n - 779 = 0. Using quadratic formula: n = (-3 + sqrt(9 - 4*2*(-779))) / 4 = (-3 + sqrt(9 + 6232)) / 4 = (-3 + 79) / 4 = 19.

AI explanation

The formula for the sum of squares of the first n natural numbers is n(n+1)(2n+1)/6. The mean is this sum divided by n, which simplifies to (n+1)(2n+1)/6. Setting the mean equal to 130 gives (n+1)(2n+1) = 780, or 2n^2 + 3n - 779 = 0. Factoring this quadratic equation or testing the options shows that n = 19 is the positive root.

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