Problem source

A frog  jumps towards a large pond.
  Each jump takes the frog either $1\,$m or $2\,$m nearer to the pond.
The probability of a $1\,$m jump is $p$ and the probability of a
$2\,$m jump is
$q$, where $p+q=1$, the occurence of long and short jumps being 
independent.
\begin{questionparts}
\item
Let $p_n(j)$ be the probability that the frog,
 starting at a point
$(n-\frac12)\,$m away from 
the edge of the pond,
 lands in the pond
for the first time on its $j$th jump.
Show that $p_2(2)=p$.
\item Let $u_n$ be the expected number of jumps,
 starting at a point
$(n-\frac12)\,$m away from 
the edge of the pond,
 required to land in the pond
for the first time. Write down the value of $u_1$.
 By finding first
the relevant values of $p_n(m)$, calculate $u_2$ and show that $u_3=
3-2q+q^2$.
\item
Given that $u_n$ can be expressed in the form $u_n= A(-q)^{n-1} +B +Cn$,
where $A$, $B$ and $C$ are constants
(independent of $n$), show that
 $C= (1+q)^{-1}$ and find $A$ and $B$ in terms of  $q$.
Hence show that, for large $n$, $u_n \approx \dfrac n{p+2q}$ and 
explain carefully why this result is to be expected.
 \end{questionparts}