<p>Let&nbsp;\(S_n\) be the sum of the first \(n\) terms of the arithmetic series \(2 +4 + 6 + \dots\) .</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Find<span class="inline-marks">[4]</span></div><ol class="i-list"><li><div class="list-item-paragraph">\(S_4\);</div></li><li><div class="list-item-paragraph">\(S_{100}\).</div></li></ol></li></ol>
<p>Let&nbsp;\(M = \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}\).</p>
<ol class="abc-list" start="2"><li><ol class="i-list"><li><div class="list-item-paragraph">Find \(M^2\).</div></li><li><div class="list-item-paragraph">Show that&nbsp;\(M^3 = \begin{pmatrix} 1 & 6 \\ 0 & 1 \end{pmatrix}\).<span class="inline-marks">[5]</span></div></li></ol></li></ol>
<p>It may now be assumed that&nbsp;\(M^n = \begin{pmatrix} 1 & 2n \\ 0 & 1 \end{pmatrix}\), for&nbsp;\(n \geq 4\). the sum of \(T_n\) is defined by</p>
<p style="text-align: center">\(T_n = M^1 + M^2 + M^3 + \dots + M^n\).</p>
<ol class="abc-list" start="3"><li><ol class="i-list"><li><div class="list-item-paragraph">Write down \(M^4\).</div></li><li><div class="list-item-paragraph">Find \(T_4\).<span class="inline-marks">[4]</span></div></li></ol></li><li><div class="list-item-paragraph">Using your results from part (a) (ii), find \(T_{100}\).<span class="inline-marks">[3]</span></div></li></ol>

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

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

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

Using your results from part (a) (ii), find \(T_{100}\).

APGP

International Baccalaureate Standard Level Mathematics