$\text{Show that}{{\text{ }}^{n}}{{C}_{r}}{{+}^{n}}{{C}_{r-1}}{{=}^{n+1}}{{C}_{r}}$

$\text{Write down the expansion of }{{(1-2x)}^{7}}$

$\text{Write down the expansion of }{{(3+x)}^{8}}\left| \text{Hint:}(3+x)=3(1+\tfrac{x}{3}) \right.$

$\begin{align}  & \text{Expand }\sqrt{9+{{x}^{2}}} \\ & \text{Giving the first four terms and stating the series is valid} \\\end{align}$

$\text{Calculate the coefficient of }{{x}^{3}}{{y}^{4}}\text{ in }{{(2x-3y)}^{7}}$

$\text{Find the term independent of }x\text{ in the expansion of }{{\left( {{x}^{2}}-\frac{2}{3x} \right)}^{9}}$

$\begin{align} & \text{Express }\frac{1}{(x-1)(x+2)}\text{ as sum of two partial fractions}\text{. Hence, write down the } \\ & \text{expansion of fraction as a series when }x\text{ is small, up to the term in }{{x}^{2}}.\text{ } \\ & \text{For values of }x\text{ is the expansion valid?} \\ \end{align}$

$\begin{align}
& \text{Show that if }x\text{ is small that }{{x}^{3}}\text{ and higher powers of }x\text{ can be neglected,} \\
& \sqrt{\frac{1+x}{1-x}}=1+x+\frac{1}{2}{{x}^{2}} \\
& \text{By putting }x=\frac{1}{7},\text{ show that }\sqrt{3}\cong \frac{196}{113} \\
\end{align}$

$\begin{align}  & \text{Find the series expansion }f(x)={{\left( \frac{2-x}{1+3x} \right)}^{\tfrac{1}{2}}}\text{ in ascending powers of }x \\ & \text{ as far as the terms in }{{x}^{2}}.\text{ For what values of }x\text{ may }f(x)\text{ be used to } \\ & \text{obtain an approximation to }\sqrt{3} \\\end{align}$