Sequence and Series - Arithmetic Progression

Let \(S_n\) be the sum of the first \(n\) terms of the arithmetic series \(2 +4 + 6 + \dots\) .

1. Find[4]
   1. \(S_4\);
   2. \(S_{100}\).

Let \(M = \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}\).

2. 1. Find \(M^2\).
   2. Show that \(M^3 = \begin{pmatrix} 1 & 6 \\ 0 & 1 \end{pmatrix}\).[5]

It may now be assumed that \(M^n = \begin{pmatrix} 1 & 2n \\ 0 & 1 \end{pmatrix}\), for \(n \geq 4\). the sum of \(T_n\) is defined by

\(T_n = M^1 + M^2 + M^3 + \dots + M^n\).

3. 1. Write down \(M^4\).
   2. Find \(T_4\).[4]
4. Using your results from part (a) (ii), find \(T_{100}\).[3]