What is the arithmetic mean of first 16 natural numbers with weights being the…

Weighted arithmetic mean:

Weighted mean = (sum of weight × value) / (sum of weights). Here each value equals its weight (both are i for i = 1..16), so numerator = sum of i^2 and denominator = sum of i.

• Numerator: sum of squares = 1^2 + 2^2 + ... + 16^2 = n(n+1)(2n+1)/6 with n = 16 = 16 × 17 × 33 / 6 = 1496.

• Denominator: sum = 1 + 2 + ... + 16 = n(n+1)/2 with n = 16 = 16 × 17 / 2 = 136.

• Weighted mean = 1496 / 136 = 11.

Answer: 11