The harmonic mean of 1, 1/2, 1/3, ..., 1/n is

Option b: 2/(n+1). Solution: Step 1: Identify the data set. The values are x1 = 1, x2 = 1/2, x3 = 1/3, ldots, xn = 1/n. Total = n terms. Step 2: Recall the HM formula. text{HM} = n/sum{i=1^{n} 1/xi} Step 3: Compute sum 1/xi. 1/xi = 1/1/i = i sum{i=1}^{n} 1/xi = sum{i=1}^{n} i = n(n+1)/2 Step 4: Calculate HM. text{HM} = n/frac{n(n+1){2}} = n x 2/n(n+1) = 2/n+1 Hence, Option B is the correct answer.

The harmonic mean of $1, 1/2, 1/3, ..., 1/n$ is

The correct answer is Option b: $2/(n+1)$. **Step 1: Identify the data set.** The values are $x_1 = 1, x_2 = \frac{1}{2}, x_3 = \frac{1}{3}, \ldots, x_n = \frac{1}{n}$. Total = $n$ terms. **Step 2: Recall the HM formula.** $$\text{HM} = \frac{n}{\sum_{i=1}^{n} \frac{1}{x_i}}$$ **Step 3: Compute $\sum \frac{1}{x_i}$.** $$\frac{1}{x_i} = \frac{1}{1/i} = i$$ $$\sum_{i=1}^{n} \frac{1}{x_i} = \sum_{i=1}^{n} i = \frac{n(n+1)}{2}$$ **Step 4: Calculate HM.** $$\text{HM} = \frac{n}{\frac{n(n+1)}{2}} = \frac{n \times 2}{n(n+1)} = \frac{2}{n+1}$$ Hence, **Option B** is the correct answer.

Central Tendency & DispersionMTP Dec 23 - Series IIQuestion 3023 of 473

All Questions

The harmonic mean of $\displaystyle 1, 1/2, 1/3, ..., 1/n$ is

Options

\displaystyle 1/(n+1)

\displaystyle 2/(n+1)

\displaystyle (n+1)/2

\displaystyle 1/(n-1)

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Correct Answer

✅ Option b — $\displaystyle 2/(n+1)$

All Options:

A $\displaystyle 1/(n+1)$
B $\displaystyle 2/(n+1)$
C $\displaystyle (n+1)/2$
D $\displaystyle 1/(n-1)$

Detailed Solution & Explanation

**Step 1: Identify the data set.** The values are

\displaystyle x_1 = 1, x_2 = \frac{1}{2}, x_3 = \frac{1}{3}, \ldots, x_n = \frac{1}{n}

. Total =

\displaystyle n

terms. **Step 2: Recall the HM formula.**

\text{HM} = \frac{n}{\sum_{i=1}^{n} \frac{1}{x_i}}

**Step 3: Compute

\displaystyle \sum \frac{1}{x_i}

.**

\frac{1}{x_i} = \frac{1}{1/i} = i

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

**Step 4: Calculate HM.**

\text{HM} = \frac{n}{\frac{n(n+1)}{2}} = \frac{n \times 2}{n(n+1)} = \frac{2}{n+1}

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

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The harmonic mean of 1,1/2,1/3,...,1/n\displaystyle 1, 1/2, 1/3, ..., 1/n1,1/2,1/3,...,1/n is