Question

If \( \frac{1}{4}, a, b, \frac{1}{19} \) form a harmonic progression (H.P.), then the values of \( a \) and \( b \) are respectively

• A. \( \frac{1}{12}, \frac{1}{15} \)
• B. \( \frac{1}{5}, \frac{1}{7} \)
• C. \( \frac{1}{9}, \frac{1}{14} \)
• D. \( \frac{1}{11}, \frac{1}{17} \)

Hint:

For harmonic progressions, take the reciprocals of the terms to convert the problem into an arithmetic progression.

Answer 1

The Correct Option is C

Step 1: Use the condition for harmonic progression.
If \( \frac{1}{4}, a, b, \frac{1}{19} \) form a harmonic progression, their reciprocals \( 4, \frac{1}{a}, \frac{1}{b}, 19 \) form an arithmetic progression (A.P.). The common difference of the A.P. is: \[ \frac{1}{a} - 4 = \frac{1}{b} - \frac{1}{a} \] Step 2: Solve for \( a \) and \( b \).
After solving this system of equations, we find that: \[ a = \frac{1}{9}, \quad b = \frac{1}{14} \] Step 3: Conclusion.
The correct answer is (C) \( \frac{1}{9}, \frac{1}{14} \).