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 Harmonic Progression (H.P.) given that the 7th term is \( \frac{1}{10} \) and the 12th term is \( \frac{1}{25} \), we can follow these steps:

### Step 1: Understand the Formula for H.P.
The nth term of a Harmonic Progression can be expressed in terms of the arithmetic progression (A.P.) of the reciprocals of the terms. The formula for the nth term of H.P. is given by:
\[ T_n = \frac{1}{A + (n-1)D} \]
where \( A \) is the first term of the corresponding A.P. and \( D \) is the common difference of the A.P.

### Step 2: Set Up the Equations for Given Terms
From the problem, we have:
- For the 7th term:
\[ T_7 = \frac{1}{A + 6D} = \frac{1}{10} \]
This leads to:
\[ A + 6D = 10 \quad \text{(Equation 1)} \]

- For the 12th term:
\[ T_{12} = \frac{1}{A + 11D} = \frac{1}{25} \]
This leads to:
\[ A + 11D = 25 \quad \text{(Equation 2)} \]

### Step 3: Solve the System of Equations
Now we have two equations:
1. \( A + 6D = 10 \)
2. \( A + 11D = 25 \)

To eliminate \( A \), we can subtract Equation 1 from Equation 2:
\[ (A + 11D) - (A + 6D) = 25 - 10 \]
This simplifies to:
\[ 5D = 15 \]
Thus, we find:
\[ D = 3 \]

### Step 4: Substitute Back to Find \( A \)
Now substitute \( D = 3 \) back into Equation 1:
\[ A + 6(3) = 10 \]
This simplifies to:
\[ A + 18 = 10 \]
Thus, we find:
\[ A = 10 - 18 = -8 \]

### Step 5: Find the 20th Term
Now we can find the 20th term using the formula:
\[ T_{20} = \frac{1}{A + 19D} \]
Substituting \( A = -8 \) and \( D = 3 \):
\[ T_{20} = \frac{1}{-8 + 19(3)} = \frac{1}{-8 + 57} = \frac{1}{49} \]

### Final Answer
Thus, the 20th term of the H.P. is:
\[ \boxed{\frac{1}{49}} \]