If `2, h_(1), h_(2),………,h_(20)6` are in harmonic progression and `2, a_(1),a_(2),……..,a_(20), 6` are in arithmetic progression, then the value of `a_(3)h_(18)` is equal to

Options

A. 6
B. 12
C. 3
D. 9

Answer

To solve the problem, we need to find the value of \( a_3 h_{18} \) given that the sequences \( 2, h_1, h_2, \ldots, h_{20}, 6 \) are in harmonic progression (HP) and \( 2, a_1, a_2, \ldots, a_{20}, 6 \) are in arithmetic progression (AP).

### Step 1: Understand the properties of HP and AP

1. **Harmonic Progression (HP)**: A sequence is in HP if the reciprocals of the terms are in arithmetic progression. Therefore, we can write:
\[ \frac{1}{2}, \frac{1}{h_1}, \frac{1}{h_2}, \ldots, \frac{1}{h_{20}}, \frac{1}{6} \]
is in AP.

2. **Arithmetic Progression (AP)**: A sequence is in AP if the difference between consecutive terms is constant. Therefore, we can write:
\[ a_n = a_1 + (n-1)d \]
where \( d \) is the common difference.

### Step 2: Find the common difference for the AP

Let’s denote the first term \( a_1 = 2 \) and the last term \( a_{21} = 6 \). The number of terms \( n = 21 \).

The common difference \( d \) can be calculated as:
\[ d = \frac{a_{21} - a_1}{n - 1} = \frac{6 - 2}{21 - 1} = \frac{4}{20} = \frac{1}{5} \]

Thus, the general term for the arithmetic progression is:
\[ a_n = 2 + (n-1) \cdot \frac{1}{5} = 2 + \frac{n-1}{5} \]

### Step 3: Find \( a_3 \)

Substituting \( n = 3 \) into the formula for \( a_n \):
\[ a_3 = 2 + \frac{3-1}{5} = 2 + \frac{2}{5} = 2 + 0.4 = 2.4 \]

### Step 4: Find the terms of the HP

From the definition of HP, we know:
\[ \frac{1}{h_n} = \frac{1}{2} + (n-1) \cdot \frac{1/6 - 1/2}{20 - 1} \]
Calculating the common difference for the reciprocals:
\[ \frac{1}{h_n} = \frac{1}{2} + (n-1) \cdot \frac{-1/3}{19} = \frac{1}{2} - \frac{(n-1)}{57} \]

### Step 5: Find \( h_{18} \)

Substituting \( n = 18 \):
\[ \frac{1}{h_{18}} = \frac{1}{2} - \frac{17}{57} \]
Calculating this:
\[ \frac{1}{h_{18}} = \frac{28.5 - 17}{57} = \frac{11.5}{57} = \frac{23}{114} \]
Thus:
\[ h_{18} = \frac{114}{23} \]

### Step 6: Calculate \( a_3 h_{18} \)

Now we can find \( a_3 h_{18} \):
\[ a_3 h_{18} = 2.4 \cdot \frac{114}{23} \]
Calculating this:
\[ = \frac{2.4 \cdot 114}{23} = \frac{273.6}{23} = 11.9 \]

### Final Answer

Thus, the value of \( a_3 h_{18} \) is \( 11.9 \).

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