\section*{5. For ALL APPLICANTS.}
This question concerns the sum $s _ { n }$ defined by

$$s _ { n } = 2 + 8 + 24 + \cdots + n 2 ^ { n }$$

(i) Let $f ( n ) = ( A n + B ) 2 ^ { n } + C$ for constants $A , B$ and $C$ yet to be determined, and suppose $s _ { n } = f ( n )$ for all $n \geqslant 1$. By setting $n = 1,2,3$, find three equations that must be satisfied by $A , B$ and $C$.\\
(ii) Solve the equations from part (i) to obtain values for $A , B$ and $C$.\\
(iii) Using these values, show that if $s _ { k } = f ( k )$ for some $k \geqslant 1$ then $s _ { k + 1 } = f ( k + 1 )$.

You may now assume that $f ( n ) = s _ { n }$ for all $n \geqslant 1$.\\
(iv) Find simplified expressions for the following sums:

$$\begin{aligned}
& t _ { n } = n + 2 ( n - 1 ) + 4 ( n - 2 ) + 8 ( n - 3 ) + \cdots + 2 ^ { n - 1 } 1 , \\
& u _ { n } = \frac { 1 } { 2 } + \frac { 2 } { 4 } + \frac { 3 } { 8 } + \cdots + \frac { n } { 2 ^ { n } } .
\end{aligned}$$

(v) Find the sum

$$\sum _ { k = 1 } ^ { n } s _ { k }$$


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