In the expansion of (9x- 1 3 x )¹⁸ , x>0 , if the term independent of x is (221)k , then k is equal to:

Step-by-step solution

The general term in the expansion of (9x - 1 3 x )¹⁸ is given by: T_ r+1 = ¹⁸C_ r (9x)^ 18-r (- 1 3 x )^r T_ r+1 = ¹⁸C_ r 9^ 18-r (- 1 3 )^r x^ 18-r x^ -r/2 T_ r+1 = ¹⁸C_ r 9^ 18-r (- 1 3 )^r x^ 18 - 3r 2 For the term independent of x , the exponent of x must be zero: 18 - 3r 2 = 0 3r 2 = 18 r = 12 Substituting r = 12 into the general term: T₁₃ = ¹⁸C₁₂ 9¹⁸⁻¹² (- 1 3 )¹² T₁₃ = ¹⁸C₁₂ 9^6 ( 1 3 )¹² T₁₃ = ¹⁸C₁₂ (3^2)^6 1 3¹² T₁₃ = ¹⁸C₁₂ 3¹² 1 3¹² = ¹⁸C₁₂ Now, calculating ¹⁸C₁₂ : ¹⁸C₁₂ = ¹⁸C₆ = 18 17 16 15 14 13 6 5 4 3