If 1 , a, b and 4 are in harmonic progression , then the value of a + b is equal to

Options

A. `5/4`
B. `10/3`
C. `3/10`
D. `4/5`

Answer

To solve the problem, we need to determine the values of \( a \) and \( b \) given that \( 1, a, b, 4 \) are in harmonic progression (HP).

### Step-by-step Solution:

1. **Understanding Harmonic Progression**:
If \( 1, a, b, 4 \) are in HP, then their reciprocals \( \frac{1}{1}, \frac{1}{a}, \frac{1}{b}, \frac{1}{4} \) will be in arithmetic progression (AP).

2. **Setting Up the Arithmetic Progression**:
The terms in AP can be expressed as:
\[ \frac{1}{1}, \frac{1}{a}, \frac{1}{b}, \frac{1}{4} \]
This means the difference between consecutive terms is constant.

3. **Finding the Common Difference**:
Let the common difference be \( d \). Then we can write:
\[ \frac{1}{a} - 1 = d \quad \text{(1)} \]
\[ \frac{1}{b} - \frac{1}{a} = d \quad \text{(2)} \]
\[ \frac{1}{4} - \frac{1}{b} = d \quad \text{(3)} \]

4. **Equating the Differences**:
From equations (1) and (2):
\[ d = \frac{1}{a} - 1 = \frac{1}{b} - \frac{1}{a} \]
Rearranging gives:
\[ \frac{1}{b} = \frac{1}{a} + d \]
Substituting \( d \) from (1):
\[ \frac{1}{b} = \frac{1}{a} + \left(\frac{1}{a} - 1\right) \]
Simplifying gives:
\[ \frac{1}{b} = \frac{2}{a} - 1 \]

5. **Substituting into the Third Equation**:
Using equation (3):
\[ d = \frac{1}{4} - \frac{1}{b} \]
Substituting \( d \) from (1):
\[ \frac{1}{a} - 1 = \frac{1}{4} - \frac{1}{b} \]
Rearranging gives:
\[ \frac{1}{b} = \frac{1}{4} - \left(\frac{1}{a} - 1\right) \]
Simplifying gives:
\[ \frac{1}{b} = \frac{1}{4} + 1 - \frac{1}{a} \]
\[ \frac{1}{b} = \frac{5}{4} - \frac{1}{a} \]

6. **Finding \( a \) and \( b \)**:
Now we have two equations:
\[ \frac{1}{b} = \frac{2}{a} - 1 \]
\[ \frac{1}{b} = \frac{5}{4} - \frac{1}{a} \]
Setting them equal:
\[ \frac{2}{a} - 1 = \frac{5}{4} - \frac{1}{a} \]
Multiplying through by \( 4a \) to eliminate the fractions:
\[ 8 - 4a = 5a - 4 \]
Rearranging gives:
\[ 8 + 4 = 5a + 4a \]
\[ 12 = 9a \implies a = \frac{4}{3} \]

7. **Finding \( b \)**:
Substitute \( a \) back into one of the equations for \( b \):
\[ \frac{1}{b} = \frac{2}{\frac{4}{3}} - 1 = \frac{3}{2} - 1 = \frac{1}{2} \]
Thus, \( b = 2 \).

8. **Calculating \( a + b \)**:
Now we can find \( a + b \):
\[ a + b = \frac{4}{3} + 2 = \frac{4}{3} + \frac{6}{3} = \frac{10}{3} \]

### Final Answer:
The value of \( a + b \) is \( \frac{10}{3} \).