If the coefficients of x^3 and x^4 in the expansion of (1+a x+b x^2 )(1-2 x)¹⁸ in powers of x are both zero, then (a, b) is equal to

Step-by-step solution

In the expansion of (1+a x+b x^2 )(1-2 x)¹⁸ coefficient of x^3 in (1+a x+b x^2 )(1-2 x)¹⁸= Coefficient of x^3 in (1-2 x)¹⁸+ coefficient of x^2 in a(1-2 x)¹⁸+ coefficient of x in b(1-2 x)¹⁸ . =- ¹⁸ C₃ 2^3+a¹⁸ C₂ 2^2-b¹⁸ C₄ 2 - ¹⁸ C₃ 2^3+a¹⁸ C₂ 2^2-b¹⁸ C₄ 2=0 18 17 16 3 2 8+a 18 17 2 2^2-b 18 2=0 17 a-b= 34 16 3 (i) Similarly, coefficient of x^4 ¹⁸ C₄ 2^4-a ¹⁸ C₃ 2^3+b ¹⁸ C₂ 2^2=0 32 a-32 b=240 (ii) From Eqs. (i) and (ii), we get a=16 and b= 272 3