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Date	May Example question	Marks available	3	Reference code	EXM.2.AHL.TZ0.7
Level	Additional Higher Level	Paper	Paper 2	Time zone	Time zone 0
Command term	Find	Question number	7	Adapted from	N/A

Question

Let $S$ $S$ _{$n$ $n$} be the sum of the first $n$ $n$ terms of the arithmetic series 2 + 4 + 6 + ….

Let M = $(\begin{matrix} 1 & 2 0 & 1 \end{matrix})$ $\left( {\begin{array}{*{20}{c}} 1&2 \\ 0&1 \end{array}} \right)$ .

It may now be assumed that M^{$n$ $n$} = $(\begin{matrix} 1 & 2 n 0 & 1 \end{matrix})$ $\left( {\begin{array}{*{20}{c}} 1&{2n} \\ 0&1 \end{array}} \right)$ , for $n$ $n$ ≥ 4. The sum T_{$n$ $n$} is defined by

T_{$n$ $n$} = M¹ + M² + M³ + ... + M^{$n$ $n$}.

Find $S$ $S$ ₄.

[1]

a.i.

Find $S$ $S$ ₁₀₀.

[3]

a.ii.

Find M².

[2]

b.i.

Show that M³ = $(\begin{matrix} 1 & 6 0 & 1 \end{matrix})$ $\left( {\begin{array}{*{20}{c}} 1&6 \\ 0&1 \end{array}} \right)$ .

[3]

b.ii.

Write down M⁴.

[1]

c.i.

Find T₄.

[3]

c.ii.

Using your results from part (a) (ii), find T₁₀₀.

[3]

Markscheme

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

$S$ $S$ ₄ = 20 A1 N1

[1 mark]

a.i.

$u$ $u$ ₁ = 2, $d$ $d$ = 2 (A1)

Attempting to use formula for $S$ $S$ _{$n$ $n$} M1

$S$ $S$ ₁₀₀ = 10100 A1 N2

[3 marks]

a.ii.

M² = $(\begin{matrix} 1 & 4 0 & 1 \end{matrix})$ $\left( {\begin{array}{*{20}{c}} 1&4 \\ 0&1 \end{array}} \right)$ A2 N2

[2 marks]

b.i.

For writing M³ as M² × M or M × M² $(or (\begin{matrix} 1 & 2 0 & 1 \end{matrix}) (\begin{matrix} 1 & 4 0 & 1 \end{matrix}))$ $\left( {{\text{or}}\left( {\begin{array}{*{20}{c}} 1&2 \\ 0&1 \end{array}} \right)\left( {\begin{array}{*{20}{c}} 1&4 \\ 0&1 \end{array}} \right)} \right)$ M1

M³ = $(\begin{matrix} 1 + 0 & 4 + 2 0 + 0 & 0 + 1 \end{matrix})$ $\left( {\begin{array}{*{20}{c}} {1 + 0}&{4 + 2} \\ {0 + 0}&{0 + 1} \end{array}} \right)$ A2

M³ = $(\begin{matrix} 1 & 6 0 & 1 \end{matrix})$ $\left( {\begin{array}{*{20}{c}} 1&6 \\ 0&1 \end{array}} \right)$ AG N0

[3 marks]

b.ii.

M⁴ = $(\begin{matrix} 1 & 8 0 & 1 \end{matrix})$ $\left( {\begin{array}{*{20}{c}} 1&8 \\ 0&1 \end{array}} \right)$ A1 N1

[1 mark]

c.i.

T₄ = $(\begin{matrix} 1 & 2 0 & 1 \end{matrix}) + (\begin{matrix} 1 & 4 0 & 1 \end{matrix}) + (\begin{matrix} 1 & 6 0 & 1 \end{matrix}) + (\begin{matrix} 1 & 8 0 & 1 \end{matrix})$ $\left( {\begin{array}{*{20}{c}} 1&2 \\ 0&1 \end{array}} \right) + \left( {\begin{array}{*{20}{c}} 1&4 \\ 0&1 \end{array}} \right) + \left( {\begin{array}{*{20}{c}} 1&6 \\ 0&1 \end{array}} \right) + \left( {\begin{array}{*{20}{c}} 1&8 \\ 0&1 \end{array}} \right)$ (M1)

= $(\begin{matrix} 4 & 20 0 & 4 \end{matrix})$ $\left( {\begin{array}{*{20}{c}} 4&{20} \\ 0&4 \end{array}} \right)$ A1A1 N3

[3 marks]

c.ii.

T₁₀₀ = $(\begin{matrix} 1 & 2 0 & 1 \end{matrix}) + (\begin{matrix} 1 & 4 0 & 1 \end{matrix}) + \dots + (\begin{matrix} 1 & 200 0 & 1 \end{matrix})$ $\left( {\begin{array}{*{20}{c}} 1&2 \\ 0&1 \end{array}} \right) + \left( {\begin{array}{*{20}{c}} 1&4 \\ 0&1 \end{array}} \right) + \ldots + \left( {\begin{array}{*{20}{c}} 1&{200} \\ 0&1 \end{array}} \right)$ (M1)