The coefficient of $$x^n$$ in the expansion of $${\left( {1 - 9x + 20{x^2}} \right)^{ - 1}}\,$$   is

Solution :
$$\eqalign{ & {\left( {1 - 9x + 20{x^2}} \right)^{ - 1}} = {\left[ {\left( {1 - 4x} \right)\left( {1 - 5x} \right)} \right]^{ - 1}} \cr & = \frac{1}{x}\left[ {\frac{{\left( {1 - 4x} \right) - \left( {1 - 5x} \right)}}{{\left( {1 - 4x} \right) \cdot \left( {1 - 5x} \right)}}} \right] = \frac{1}{x}\left[ {{{\left( {1 - 5x} \right)}^{ - 1}} - {{\left( {1 - 4x} \right)}^{ - 1}}} \right] \cr & = \frac{1}{5}\left[ {\left( {5 - 4} \right)x + \left( {{5^2} - {4^2}} \right){x^2} + \left( {{5^3} - {4^3}} \right){x^3} + ..... + \left( {{5^n} - {4^n}} \right){x^n} + .....} \right] \cr & \therefore {\text{coeff}}{\text{. of }}{x^n} = {5^{n + 1}} - {4^{n + 1}} \cr} $$