A sequence \(U _ { 1 } , U _ { 2 } , U _ { 3 } , \ldots\) is defined by
$$\begin{gathered}
U _ { n + 2 } = 2 U _ { n + 1 } - U _ { n } , \quad n \geqslant 1
U _ { 1 } = 4 \text { and } U _ { 2 } = 4
\end{gathered}$$
Find the value of
(a) \(U _ { 3 }\)
(b) \(\sum _ { n = 1 } ^ { 20 } U _ { n }\)
Another sequence \(V _ { 1 } , V _ { 2 } , V _ { 3 } , \ldots\) is defined by
(a) Find \(V _ { 3 }\) and \(V _ { 4 }\) in terms of \(k\).
$$\begin{gathered}
V _ { n + 2 } = 2 V _ { n + 1 } - V _ { n } , \quad n \geqslant 1
V _ { 1 } = k \text { and } V _ { 2 } = 2 k , \text { where } k \text { is a constant }
\end{gathered}$$
a) Find \(V _ { 3 }\)