If A.M., G.M., and H.M. of the first and last terms of the series of `100 , 101 , 102 ,...n-1,n` are the terms of the series itself, then the value of `ni s(100

Options

A. 200
B. 300
C. 400
D. 500

Explanation

To solve the problem, we need to find the value of \( n \) such that the Arithmetic Mean (A.M.), Geometric Mean (G.M.), and Harmonic Mean (H.M.) of the first term (100) and the last term (\( n \)) of the series \( 100, 101, 102, \ldots, n \) are also terms of the series itself.

### Step-by-Step Solution:

1. **Identify the First and Last Terms:**
The first term \( a = 100 \) and the last term \( b = n \).

2. **Calculate the Arithmetic Mean (A.M.):**
\[ A.M. = \frac{a + b}{2} = \frac{100 + n}{2} \]

3. **Calculate the Geometric Mean (G.M.):**
\[ G.M. = \sqrt{a \cdot b} = \sqrt{100 \cdot n} = 10\sqrt{n} \]

4. **Calculate the Harmonic Mean (H.M.):**
\[ H.M. = \frac{2ab}{a + b} = \frac{2 \cdot 100 \cdot n}{100 + n} = \frac{200n}{100 + n} \]

5. **Set Conditions for A.M., G.M., and H.M. to be Terms of the Series:**
Since \( A.M. \), \( G.M. \), and \( H.M. \) must be integers and also terms of the series:
- \( \frac{100 + n}{2} \) must be an integer, implying \( n \) must be even.
- \( 10\sqrt{n} \) must be an integer, implying \( n \) must be a perfect square.
- \( \frac{200n}{100 + n} \) must be an integer.

6. **Let \( n = k^2 \) (since \( n \) is a perfect square):**
- Since \( n \) is even, let \( n = 2m^2 \) for some integer \( m \).

7. **Substitute \( n \) into H.M. Condition:**
\[ H.M. = \frac{200(2m^2)}{100 + 2m^2} = \frac{400m^2}{100 + 2m^2} \]
For \( H.M. \) to be an integer, \( 100 + 2m^2 \) must divide \( 400m^2 \).

8. **Simplify the Condition:**
We can rewrite the condition:
\[ 400m^2 \div (100 + 2m^2) \]
This means \( 100 + 2m^2 \) must divide \( 400m^2 \).

9. **Check Possible Values of \( n \):**
Since \( n \) must be between 100 and 500, we check perfect squares that are even:
- \( 100, 144, 196, 256, 324, 400, 484 \)

10. **Evaluate Each Perfect Square:**
- \( n = 100 \): \( H.M. = 200 \) (not a term)
- \( n = 144 \): \( H.M. = 96 \) (not a term)
- \( n = 196 \): \( H.M. = 78 \) (not a term)
- \( n = 256 \): \( H.M. = 64 \) (not a term)
- \( n = 324 \): \( H.M. = 54 \) (not a term)
- \( n = 400 \): \( H.M. = 50 \) (is a term)
- \( n = 484 \): \( H.M. = 48 \) (not a term)

11. **Conclusion:**
The only value of \( n \) that satisfies all conditions is \( n = 400 \).

### Final Answer:
\[ \boxed{400} \]