Find the coefficient of x18 in (x2 - \(\frac{6}{x}\))15

1.	Find the coefficient of x18 in (x2 - \(\frac{6}{x}\))15
Answer» Here, x → x² , a → \(\frac{-6}{x}\) and r = 15 T_r+1 = ¹⁵C_r.(x²)^15-r(\(\frac{-6}{x}\))^r T_r+1 = ¹⁵C_r.x^30-2r.(-6)^r .x^-r = ¹⁵C_r.(-6)^r .x^30-2r-r = ¹⁵C_r(-6)^r.x^30-3r . To find the coefficient of x¹⁸,equate the power of x to 18 ∴ 30 – 3r = 18 ⇒ 30 – 18 = 3r ⇒ 3r = 12 ⇒ r = 4 T₄₊₁ = ¹⁵C₄(-6) ⁴x¹⁸ ∴ Coefficient of x¹⁸ is ¹⁵C₄. (6)⁴

Answer»

Here, x → x² , a → \(\frac{-6}{x}\) and r = 15

T_r+1 = ¹⁵C_r.(x²)^15-r(\(\frac{-6}{x}\))^r

T_r+1 = ¹⁵C_r.x^30-2r.(-6)^r .x^-r

= ¹⁵C_r.(-6)^r .x^30-2r-r

= ¹⁵C_r(-6)^r.x^30-3r .

To find the coefficient of x¹⁸,equate the power of x to 18

∴ 30 – 3r = 18

⇒ 30 – 18 = 3r

⇒ 3r = 12

⇒ r = 4

T₄₊₁ = ¹⁵C₄(-6) ⁴x¹⁸

∴ Coefficient of x¹⁸ is ¹⁵C₄. (6)⁴

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