Given that the first two terms, in ascending powers of \(x\), in the series expansion of \(\mathrm{f}(x)\) are 384 and \(-104 x\)

Question

(a) Find the first 3 terms in ascending powers of \(x\) of the binomial expansion of \(\left(2-\frac{x}{8}\right)^{7}\) \(\mathrm{f}(x)=(a x+b)\left(2-\frac{x}{8}\right)^{7}\) where \(a\) and \(b\) are constants
Given that the first two terms, in ascending powers of \(x\), in the series expansion of \(\mathrm{f}(x)\) are 384 and \(-104 x\)

(b) Find the values of \(a\) and \(b\)

Answer 1

(a)

\(\left(2-\frac{x}{8}\right)^{7}\)
\(1(2)^{7}+7(2)^{6}\left(-\frac{x}{3}\right)+21(2)^{5}\left(-\frac{x}{8}\right)^{2}\)
\(128-56 x+\frac{21}{2} x^{2}\)

(b)

\((a x+b)\left(128-56 x+\frac{21}{2} x^{2}\right)\)
\(128 a x+128 b-56 a x^{2}-56 b x \ldots\)
\(128 b+(128 a-56 b) x \ldots\)
\(128 b=384 \quad 128 a-56(3)=-104\)
\(b=3\)
\(128 a=64\)
\(a=\frac{1}{2}\)