1.10: Past-paper question 6

(a) Show that

The expansion of $\displaystyle \frac{a}{\sqrt{4+bx}}$ in ascending powers of $\displaystyle x$ can be written as

where $\displaystyle P$, $\displaystyle Q$, $\displaystyle R$ and $\displaystyle S$ are rational numbers.

(b) Show that $\displaystyle Q = -\frac{ab}{16}$ and $\displaystyle S = -\frac{5ab^3}{2048}$

and find $\displaystyle P$ and $\displaystyle R$ in terms of $\displaystyle a$ and $\displaystyle b$, as fractions in their lowest terms.

Given that $\displaystyle Q = \frac{128}{5}S$ and $\displaystyle R = \frac{9}{256}$

(c) show that $\displaystyle a = 3$ and $\displaystyle b = 1$

(d) Hence, using an appropriate value of $\displaystyle x$, find, to 3 decimal places, an approximate

value for $\displaystyle \frac{\sqrt{6}}{2}$