Given the weights for the numbers 1, 2, 3, ..., n are respectively 12, 22, 32, ..., n2 then weighted HM is

Option c: 2n+1/3. Solution: Step 1: Recall the formula for Weighted Harmonic Mean. Weighted HM = sum wi/sum where xi = i and wi = i2 . Step 2: Compute numerator. sumi=1n wi = sumi=1n i2 = n(n+1)(2n+1)/6 Step 3: Compute denominator. sumi=1n wi/xi = sumi=1n i2/i = sumi=1n i = n(n+1)/2 Step 4: Calculate Weighted HM. WHM = (n+1)(2n+1)/6(n+1)2 = n(n+1)(2n+1)/6 × 2/n(n+1) = 2n+1/3 Hence, Option C is the correct answer.

Measures of Central Tendency and DispersionMCQPYQ Nov. 20Question 3000 of 473

All Questions

Given the weights for the numbers $\displaystyle 1, 2, 3, \dots, n$ are respectively $\displaystyle 1^2, 2^2, 3^2, \dots, n^2$ then weighted HM is

Options

\displaystyle \frac{2n+1}{4}

\displaystyle \frac{2n+1}{6}

\displaystyle \frac{2n+1}{3}

\displaystyle \frac{2n+1}{2}

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Correct Answer

✅ Option c — $\displaystyle \frac{2n+1}{3}$

All Options:

A $\displaystyle \frac{2n+1}{4}$
B $\displaystyle \frac{2n+1}{6}$
C $\displaystyle \frac{2n+1}{3}$
D $\displaystyle \frac{2n+1}{2}$

Detailed Solution & Explanation

**Step 1: Recall the formula for Weighted Harmonic Mean.**

\text{Weighted HM} = \frac{\sum w_i}{\sum \frac{w_i}{x_i}}

where

\displaystyle x_i = i

and

\displaystyle w_i = i^2

. **Step 2: Compute numerator.**

\sum_{i=1}^{n} w_i = \sum_{i=1}^{n} i^2 = \frac{n(n+1)(2n+1)}{6}

**Step 3: Compute denominator.**

\sum_{i=1}^{n} \frac{w_i}{x_i} = \sum_{i=1}^{n} \frac{i^2}{i} = \sum_{i=1}^{n} i = \frac{n(n+1)}{2}

**Step 4: Calculate Weighted HM.**

\text{WHM} = \frac{\dfrac{n(n+1)(2n+1)}{6}}{\dfrac{n(n+1)}{2}} = \frac{n(n+1)(2n+1)}{6} \times \frac{2}{n(n+1)} = \frac{2n+1}{3}

Hence, **Option C** is the correct answer.

About This Chapter: Measures of Central Tendency and Dispersion

Paper

Paper 3: Quantitative Aptitude

Weightage

12-15 Marks

Key Topics

Mean, Median, Mode, Range, Mean Deviation, Standard Deviation

The core foundation of Statistics. This chapter covers Mean (Arithmetic, Geometric, Harmonic), Median, Mode, and their properties. It also explores measures of spread like Range, Mean Deviation, Standard Deviation, and Quartile Deviation.

View Official ICAI Syllabus

Exam Strategy Tip

Do not just memorize formulas; ICAI loves asking about the mathematical properties (e.g., 'sum of deviations from the AM is always zero'). You can usually eliminate 2 options just by knowing the properties.

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Given the weights for the numbers 1,2,3,…,n\displaystyle 1, 2, 3, \dots, n1,2,3,…,n are respectively 12,22,32,…,n2\displaystyle 1^2, 2^2, 3^2, \dots, n^212,22,32,…,n2 then weighted HM is