If the `7^(th)` terms of a H.P. is 8 and the `8^(th)` term is 7. Then find the `28^(th)` term

Answer

To solve the problem step-by-step, we will use the properties of Harmonic Progression (H.P.) and its relationship with Arithmetic Progression (A.P.).

### Step 1: Understand the relationship between H.P. and A.P.
In a Harmonic Progression, if the terms are \( A, B, C \), then the reciprocals \( \frac{1}{A}, \frac{1}{B}, \frac{1}{C} \) are in Arithmetic Progression.

### Step 2: Define the terms given in the problem.
We are given:
- The 7th term of the H.P. is 8, which means \( T_7 = 8 \).
- The 8th term of the H.P. is 7, which means \( T_8 = 7 \).

### Step 3: Convert H.P. terms to A.P. terms.
From the properties of H.P.:
- The 7th term in A.P. corresponding to the 7th term in H.P. is \( \frac{1}{T_7} = \frac{1}{8} \).
- The 8th term in A.P. corresponding to the 8th term in H.P. is \( \frac{1}{T_8} = \frac{1}{7} \).

### Step 4: Set up equations for the A.P. terms.
Let \( A \) be the first term and \( D \) be the common difference of the A.P. Then we can write:
1. \( A + 6D = \frac{1}{8} \) (for the 7th term)
2. \( A + 7D = \frac{1}{7} \) (for the 8th term)

### Step 5: Solve the equations.
We can subtract the first equation from the second:
\[ (A + 7D) - (A + 6D) = \frac{1}{7} - \frac{1}{8} \]
This simplifies to:
\[ D = \frac{1}{7} - \frac{1}{8} \]
Finding a common denominator (56):
\[ D = \frac{8 - 7}{56} = \frac{1}{56} \]

### Step 6: Substitute \( D \) back to find \( A \).
Now substitute \( D \) back into one of the equations to find \( A \):
\[ A + 6 \left(\frac{1}{56}\right) = \frac{1}{8} \]
This simplifies to:
\[ A + \frac{6}{56} = \frac{1}{8} \]
Convert \( \frac{1}{8} \) to have a denominator of 56:
\[ A + \frac{3}{28} = \frac{7}{56} \]
Now, subtract \( \frac{3}{28} \) from both sides:
\[ A = \frac{7}{56} - \frac{6}{56} = \frac{1}{56} \]

### Step 7: Find the 28th term of the A.P.
The formula for the \( n^{th} \) term of an A.P. is:
\[ T_n = A + (n-1)D \]
For the 28th term:
\[ T_{28} = A + 27D = \frac{1}{56} + 27 \left(\frac{1}{56}\right) = \frac{1 + 27}{56} = \frac{28}{56} = \frac{1}{2} \]

### Step 8: Convert back to H.P.
The 28th term of the H.P. is the reciprocal of the 28th term of the A.P.:
\[ T_{28}^{HP} = \frac{1}{T_{28}^{AP}} = \frac{1}{\frac{1}{2}} = 2 \]

### Final Answer:
The 28th term of the Harmonic Progression is \( 2 \).