If the 4th and the 7th terms of an H.P. are `1/2 and 2/7` respectively . Find the first term.

Answer

To find the first term of the Harmonic Progression (H.P.) given that the 4th term is \( \frac{1}{2} \) and the 7th term is \( \frac{2}{7} \), we can 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.). Therefore, if the 4th term of the H.P. is \( \frac{1}{2} \), the corresponding term in the A.P. will be \( 2 \) (since \( \frac{1}{\frac{1}{2}} = 2 \)). Similarly, if the 7th term of the H.P. is \( \frac{2}{7} \), the corresponding term in the A.P. will be \( \frac{7}{2} \).

### Step 2: Set up equations for the A.P.
Let \( a \) be the first term and \( d \) be the common difference of the A.P. The terms can be expressed as:
- 4th term of A.P.: \( a + 3d = 2 \) (from the 4th term of H.P.)
- 7th term of A.P.: \( a + 6d = \frac{7}{2} \) (from the 7th term of H.P.)

### Step 3: Write down the equations
We have the following two equations:
1. \( a + 3d = 2 \) (Equation 1)
2. \( a + 6d = \frac{7}{2} \) (Equation 2)

### Step 4: Subtract the equations
To eliminate \( a \), we can subtract Equation 1 from Equation 2:
\[ (a + 6d) - (a + 3d) = \frac{7}{2} - 2 \]
This simplifies to:
\[ 3d = \frac{7}{2} - 2 \]

### Step 5: Simplify the right side
Convert \( 2 \) to a fraction with a common denominator:
\[ 2 = \frac{4}{2} \]
Thus:
\[ 3d = \frac{7}{2} - \frac{4}{2} = \frac{3}{2} \]

### Step 6: Solve for \( d \)
Now, divide both sides by 3:
\[ d = \frac{3}{2} \times \frac{1}{3} = \frac{1}{2} \]

### Step 7: Substitute \( d \) back to find \( a \)
Now substitute \( d = \frac{1}{2} \) back into Equation 1:
\[ a + 3 \left(\frac{1}{2}\right) = 2 \]
This simplifies to:
\[ a + \frac{3}{2} = 2 \]
Subtract \( \frac{3}{2} \) from both sides:
\[ a = 2 - \frac{3}{2} = \frac{4}{2} - \frac{3}{2} = \frac{1}{2} \]

### Step 8: Conclusion
Thus, the first term \( a \) of the H.P. is \( \frac{1}{2} \).