If the first two terms of a H.P. are `2//5and12//23` respectively. Then, largest term is

Explanation

To solve the problem step by step, we will follow the process of finding the terms of the Harmonic Progression (H.P.) and determining the largest term.

### Step 1: Understanding the terms of H.P.
The first two terms of the H.P. are given as \( \frac{2}{5} \) and \( \frac{12}{23} \). In a Harmonic Progression, the reciprocals of the terms form an Arithmetic Progression (A.P.).

### Step 2: Finding the first two terms of A.P.
Let the first term of the corresponding A.P. be \( a \) and the common difference be \( d \). Therefore, we have:
- First term: \( \frac{1}{\frac{2}{5}} = \frac{5}{2} \)
- Second term: \( \frac{1}{\frac{12}{23}} = \frac{23}{12} \)

### Step 3: Setting up the equations
From the above, we can set up the following equations:
1. \( a = \frac{5}{2} \)
2. \( a + d = \frac{23}{12} \)

### Step 4: Solving for \( d \)
Substituting \( a \) into the second equation:
\[ \frac{5}{2} + d = \frac{23}{12} \]
To solve for \( d \), we first convert \( \frac{5}{2} \) to have a common denominator of 12:
\[ \frac{5}{2} = \frac{30}{12} \]
Now substituting:
\[ \frac{30}{12} + d = \frac{23}{12} \]
Subtracting \( \frac{30}{12} \) from both sides gives:
\[ d = \frac{23}{12} - \frac{30}{12} = -\frac{7}{12} \]

### Step 5: Finding the nth term of H.P.
The nth term of the H.P. can be expressed as:
\[ H_n = \frac{1}{a + (n-1)d} \]
Substituting the values of \( a \) and \( d \):
\[ H_n = \frac{1}{\frac{5}{2} + (n-1)(-\frac{7}{12})} \]
This simplifies to:
\[ H_n = \frac{1}{\frac{5}{2} - \frac{7(n-1)}{12}} \]

### Step 6: Finding a common denominator
To simplify further, convert \( \frac{5}{2} \) to a fraction with a denominator of 12:
\[ \frac{5}{2} = \frac{30}{12} \]
Thus,
\[ H_n = \frac{1}{\frac{30 - 7(n-1)}{12}} = \frac{12}{30 - 7(n-1)} \]
This can be rewritten as:
\[ H_n = \frac{12}{37 - 7n} \]

### Step 7: Finding the largest term
To find the largest term, we need to minimize the denominator \( 37 - 7n \) while ensuring it remains positive:
\[ 37 - 7n > 0 \implies n < \frac{37}{7} \approx 5.2857 \]
Thus, the maximum integer \( n \) can take is 5.

### Step 8: Calculating the 5th term
Substituting \( n = 5 \):
\[ H_5 = \frac{12}{37 - 7 \times 5} = \frac{12}{37 - 35} = \frac{12}{2} = 6 \]

### Conclusion
The largest term in the given H.P. is \( 6 \).

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