In the expansion in powers of x of the function
\(\displaystyle (1+x)(a-bx)^{12}\)
the coefficient of the x-cubed term is zero. Find in its simplest form the value of the ratio a/b.
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+397 \(\displaystyle (1+x)(a-bx)^{12} = \\ \displaystyle (1+x)\{(^{12}_{\, 0})a^{12}+( ^{12}_{\, 1})a^{11}(-bx)+( ^{12}_{ \,2})a^{10}(-bx)^{2}+( ^{12}_{ \, 3})a^{9}(-bx)^{3}+\dots \} \\ \displaystyle = (1+x)(a^{12}-12a^{11}bx+66a^{10}b^{2}x^{2}-220a^{9}b^{3}x^{3}+ \dots) \\ \displaystyle = a^{12}+(a^{12}-12a^{11}b)x-(12a^{11}b-66a^{10}b^{2})x^{2}+(66a^{10}b^{2}-220a^{9}b^{3})x^{3}+\dots \)
If the coefficient of the x-cubed term is zero, then
\(\displaystyle 66a^{10}b^{2}=220a^{9}b^{3} \\ \displaystyle \frac{a}{b}=\frac{220}{66}=\frac{10}{3}.\)
