1.3: Binomial expansion and approximation

(a) Write

in the form $\displaystyle a(1-bx)^{-3}$, where $\displaystyle a$ and $\displaystyle b$ are fractions in their lowest terms.

(b) Expand $\displaystyle \frac{3}{(3-x)^3}$ in ascending powers of $\displaystyle x$ up to and including the term in $\displaystyle x^3$. Express each coefficient as a fraction in its lowest terms.

(c) (i) Use a suitable value of $\displaystyle x$ with your expansion in part (b), to obtain an approximation for $\displaystyle \frac{24}{125}$ to 5 decimal places.

(ii) Find the percentage error, to 2 decimal places, of your approximation from the actual value.