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Given an infinite sequence  of numbers
$u_0$, $u_1$, $u_2$, $\ldots\,$, we define the
{\em generating function}, $\f$, for the sequence by
\[
\f(x) =  u_0 + u_1x +u_2 x^2 +u_3 x^3 + \cdots \,.
\]
Issues of convergence can be ignored in this question.

\begin{questionparts}

\item 
Using the binomial series, show that 
the sequence given by $u_n=n\,$ has generating function
$x(1-x)^{-2}$,
and
find the sequence that has generating function $x(1-x)^{-3}$.

Hence, or otherwise, find the generating function for the 
sequence $u_n =n^2$. You should simplify your answer.

\item 
\begin{itemize}
\item[\bf (a)]
The sequence $u_0$, $u_1$, $u_2$, $\ldots\,$ is 
 determined by
$u_{n} = ku_{n-1}$ ($n\ge1$),
where $k$ is independent of $n$,
and $u_0=a$. 
By summing 
the identity $u_{n}x^n \equiv ku_{n-1}x^n$, or otherwise,
show that
 the generating function, f, satisfies
\[
\f(x) = a + kx \f(x)
\,.
\]
Write down  an expression for $\f(x)$.

\vspace{3mm}
\item[\bf (b)]
The sequence 
 $u_0$, $u_1$, $u_2$, $\ldots\,$ is 
 determined by
$u_{n} = u_{n-1}+ u_{n-2}$ ($n\ge2$)
and $u_0=0$, $u_1=1$. 
Obtain the generating function.

\end{itemize}
\end{questionparts}

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