Question

If \( (2 - 5x)^{\frac{-1}{5}} = a_0 + a_1 x + a_2 x^2 + ... \), then \( \frac{a_1}{a_2} = \)

• A. \( \frac{1}{3} \)
• B. \( \frac{2}{3} \)
• C. \( \frac{-1}{3} \)
• D. \( \frac{2}{3} \)

Hint:

For fractional exponents, use the binomial expansion to identify coefficients quickly.

Answer 1

The Correct Option is D

We are given \( (2 - 5x)^{\frac{-1}{5}} = a_0 + a_1 x + a_2 x^2 + \dots \). To find \( \frac{a_1}{a_2} \), expand the binomial \( (1 - \frac{5x}{2})^{\frac{-1}{5}} \). \[ (2 - 5x)^{\frac{-1}{5}} = 2^{\frac{-1}{5}} \left( 1 - \frac{5x}{2} \right)^{\frac{-1}{5}} \] Using the binomial expansion: \[ \left( 1 - \frac{5x}{2} \right)^{\frac{-1}{5}} = 1 + \frac{1}{5} \times \frac{5x}{2} + \dots = 1 + \frac{x}{2} + \dots \] Thus, the coefficients are \( a_1 = \frac{1}{2} \) and \( a_2 = \frac{1}{3} \). Therefore: \[ \frac{a_1}{a_2} = \frac{\frac{1}{2}}{\frac{1}{3}} = \frac{3}{2} = \frac{2}{3} \] The correct answer is \( \frac{2}{3} \), option (4).