The 7th term of a H.P. is `(1)/(10)` and 12 th term is `(1)/(25),` find the 20th term of H.P.

Answer

To find the 20th term of the given Harmonic Progression (H.P.), we will follow these steps:

### Step 1: Understand the relationship between H.P. and A.P.
The terms of a Harmonic Progression can be expressed as the reciprocals of the terms of an Arithmetic Progression (A.P.). If the terms of the H.P. are \( T_n \), then the corresponding terms of the A.P. can be denoted as \( A_n \).

### Step 2: Write down the given terms
We know:
- The 7th term of the H.P. is \( T_7 = \frac{1}{10} \)
- The 12th term of the H.P. is \( T_{12} = \frac{1}{25} \)

### Step 3: Find the corresponding terms in A.P.
The corresponding terms in the A.P. would be:
- \( A_7 = \frac{1}{T_7} = 10 \)
- \( A_{12} = \frac{1}{T_{12}} = 25 \)

### Step 4: Write the general term of the A.P.
The general term of an A.P. can be expressed as:
\[ A_n = A + (n-1)D \]
where \( A \) is the first term and \( D \) is the common difference.

### Step 5: Set up equations for the given terms
From the information we have:
1. For the 7th term:
\[ A + 6D = 10 \quad \text{(1)} \]
2. For the 12th term:
\[ A + 11D = 25 \quad \text{(2)} \]

### Step 6: Solve the equations
Subtract equation (1) from equation (2):
\[ (A + 11D) - (A + 6D) = 25 - 10 \]
This simplifies to:
\[ 5D = 15 \]
Thus, we find:
\[ D = 3 \]

### Step 7: Substitute \( D \) back to find \( A \)
Substituting \( D = 3 \) back into equation (1):
\[ A + 6(3) = 10 \]
\[ A + 18 = 10 \]
\[ A = 10 - 18 = -8 \]

### Step 8: Find the 20th term of the A.P.
Now we can find the 20th term of the A.P. using:
\[ A_{20} = A + 19D \]
Substituting \( A = -8 \) and \( D = 3 \):
\[ A_{20} = -8 + 19(3) = -8 + 57 = 49 \]

### Step 9: Find the 20th term of the H.P.
The 20th term of the H.P. is the reciprocal of the 20th term of the A.P.:
\[ T_{20} = \frac{1}{A_{20}} = \frac{1}{49} \]

### Final Answer:
The 20th term of the H.P. is \( \frac{1}{49} \).

---