arrow_back Find the first 3 terms in ascending powers of \(x\) of the binomial expansion of \(\left(2+\frac{x}{2}\right)^{6}\)

Question

(a) Find the first 3 terms in ascending powers of \(x\) of the binomial expansion of \(\left(2+\frac{x}{2}\right)^{6}\)

(b) Use your expansion to find an estimate for the value of \(2.05^{6}\)

Answer 1

(a)

\(1(2)^{6}+6(2)^{5}\left(\frac{x}{2}\right)+15(2)^{4}\left(\frac{x}{2}\right)^{2}\)
\(= 64+96 x+60 x^{2}\)

(b)

\(\begin{aligned}\left(2+\frac{x}{2}\right)^{6} &=2.05^{6} \\ 2+\frac{x}{2} &=2.05 \\ \frac{x}{2} &=0.05 \\ x &=0.1 \\=74.2 & \\=& \\=& \\=& \\=& \\ 24.1 &+60(0.1)^{2} \end{aligned}\)