What is the arithmetic mean of first 16 natural numbers with weights being the…
2025
What is the arithmetic mean of first 16 natural numbers with weights being the number itself?
- A.
11
- B.
16/7
- C.
19/3
- D.
39/4
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Show answer & explanation
Correct answer: A
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