Official solution
\[{1 \over {\left( {1 - ax} \right)\left( {1 - bx} \right)}}\]
= \[{\left( {1 - ax} \right)^{ - 1}}{\left( {1 - bx} \right)^{ - 1}}\]
= \[\left[ {1 + \left( { - 1} \right)\left( { - ax} \right) + {{\left( { - 1} \right)\left( { - 2} \right)} \over {1.2}}{{\left( { - ax} \right)}^2} + ...} \right]\] -
\[\left[ {1 + \left( { - 1} \right)\left( { - bx} \right) + {{\left( { - 1} \right)\left( { - 2} \right)} \over {1.2}}{{\left( { - bx} \right)}^2} + ...} \right]\]
= \[\left[ {1 + ax + {a^2}{x^2} + ... + {a^{n - 1}}{x^{n - 1}} + {a^n}{x^n}}+.... \right]\] -
\[\left[ {1 + bx + {b^2}{x^2} + ... + {b^{n - 1}}{x^{n - 1}} + {b^n}{x^n}}+.... \right]\]
Coefficient of xn =
\[{a^n} + {a^{n - 1}}b + {a^{n - 2}}{b^2} + .... + {b^n}\]
= \[{a^n}\left[ {1 + {b \over a} + {{{b^2}} \over {{a^2}}} + ..... + {{{b^n}} \over {{a^n}}}} \right]\]
= \[{a^n}\left[ {{{{{\left( {{b \over a}} \right)}^{n + 1}} - 1} \over {{b \over a} - 1}}} \right]\]
= \[{a^n}\left[ {{{{b^{n + 1}} - {a^{n + 1}}} \over {{a^{n + 1}}\left( {{{b - a} \over a}} \right)}}} \right]\]
= \[{{{{b^{n + 1}} - {a^{n + 1}}} \over {b - a}}}\]