If `a_(1), a_(2), a_(3) and h_(1), h_(2), h_(3)` are the A.M.'s and H.M.'s between 2 and 3, then `a_(2) h_(2)` is equal to

Options

A. 1
B. 4
C. 6
D. 8

Answer

To solve the problem, we need to find the values of the second arithmetic mean \( a_2 \) and the second harmonic mean \( h_2 \) between the numbers 2 and 3, and then compute the product \( a_2 h_2 \).

### Step-by-Step Solution:

1. **Identify the Arithmetic Means (A.M.)**:
The arithmetic means \( a_1, a_2, a_3 \) between 2 and 3 can be calculated using the formula for the \( n \)-th term of an arithmetic progression (AP):
\[ a_n = a + (n-1)d \]
where \( a \) is the first term, \( d \) is the common difference, and \( n \) is the term number.

2. **Calculate the Common Difference (d)**:
The first term \( a = 2 \) and the last term (3) can be expressed as:
\[ a_1 = 2, \quad a_4 = 3 \]
Since there are 3 A.M.s, there are 4 terms in total (2, \( a_1, a_2, a_3, 3 \)). The common difference \( d \) can be calculated as:
\[ d = \frac{3 - 2}{4 - 1} = \frac{1}{3} \]

3. **Find the Second Arithmetic Mean (a_2)**:
Now, we can find \( a_2 \):
\[ a_2 = a + (2)d = 2 + (2)\left(\frac{1}{3}\right) = 2 + \frac{2}{3} = \frac{6}{3} + \frac{2}{3} = \frac{8}{3} \]

4. **Identify the Harmonic Means (H.M.)**:
The harmonic means \( h_1, h_2, h_3 \) can be calculated using the relationship between A.M. and H.M.:
\[ h_n = \frac{2ab}{a + b} \]
where \( a \) and \( b \) are the terms between which we are finding the H.M.

5. **Calculate the Second Harmonic Mean (h_2)**:
The second harmonic mean \( h_2 \) can be calculated as:
\[ h_2 = \frac{2 \cdot 2 \cdot 3}{2 + 3} = \frac{12}{5} \]

6. **Calculate the Product \( a_2 h_2 \)**:
Now we can find the product:
\[ a_2 h_2 = \left(\frac{8}{3}\right) \left(\frac{12}{5}\right) = \frac{96}{15} = \frac{32}{5} \]

### Final Answer:
Thus, \( a_2 h_2 = \frac{32}{5} \).