Multiple choice

N observations on a variable x are $x_i=A+iB$ for $i=1, 2, 3,..., n$ where A, B are real constants. The mean of the observation is?

  1. $A+B\dfrac{(n+1)}{2}$
  2. $nA+B\dfrac{(n+1)}{2}$
  3. $A+Bn\dfrac{(n+1)}{2}$
  4. $A+B\left(\dfrac{n}{2}\right)$
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A Correct answer
Explanation

The mean is the sum of all observations divided by n. The sum is sum(A + iB) from i=1 to n, which equals nA + B * (n * (n + 1) / 2). Dividing by n gives A + B * (n + 1) / 2.

AI explanation

The mean of the N observations is calculated as the sum of (A + iB) for i from 1 to n, divided by n. This can be split into the sum of A divided by n plus B times the sum of i divided by n, which simplifies to A plus B times the sum of the first n natural numbers divided by n. Using the formula for the sum of the first n natural numbers, n(n+1)/2, the mean becomes A + B*(n+1)/2.

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