Higher June 2024 Paper 1 Q23

(a) The first three terms of a geometric progression are \(\quad \dfrac{\sqrt{5}}{2} \qquad \dfrac{5}{4} \qquad \dfrac{5\sqrt{5}}{8}\)

Work out the next term. [1 mark]

(b) The \(n\)th term of a sequence is \(\quad (2 + \sqrt{3})^n\)

Show that the third term is \(\quad 26 + 15\sqrt{3}\) [3 marks]

Mark scheme (a)

Answer	Mark	Comments
\(\dfrac{25}{16}\) or \(1\dfrac{9}{16}\)	B1	oe with no surds or indices

Ignore an incorrect conversion of \(\dfrac{25}{16}\) to a mixed number
\(\dfrac{5\sqrt{5}\sqrt{5}}{16}\) or \(\dfrac{5^2}{16}\)	B0

Mark scheme (b)

Answer	Mark	Comments
\(4 + 2\sqrt{3} + 2\sqrt{3} + (\sqrt{3})^2\) or \(4 + 4\sqrt{3} + (\sqrt{3})^2\) or \(7 + 4\sqrt{3}\)	M1	oe 4 terms with at least 3 correct or 3 terms with 2 correct including \(4\sqrt{3}\) terms may be seen in a grid
\(7 \times 2 + 7\sqrt{3} + 2 \times 4\sqrt{3} + 4\sqrt{3} \times \sqrt{3}\) or \(8 + 8\sqrt{3} + 6 + 4\sqrt{3} + 4 \times 3 + 3\sqrt{3}\) or \(14 + 7\sqrt{3} + 8\sqrt{3} + 12\) or \(8 + 4\sqrt{3} + 4\sqrt{3} + 6 + 4\sqrt{3} + 6 + 6 + 3\sqrt{3}\)	M1dep	oe full expansion with correct multiplication of their 2, 3 or 4 terms by \((2 + \sqrt{3})\) terms may be seen in a grid
\(8 + 4\sqrt{3} + 4\sqrt{3} + 6 + 4\sqrt{3} + 6 + 6 + 3\sqrt{3}\) and \(26 + 15\sqrt{3}\) or \(14 + 7\sqrt{3} + 8\sqrt{3} + 12\) and \(26 + 15\sqrt{3}\)	A1	oe with full expansion terms may be seen in a grid condone \(15\sqrt{3} + 26\)

Remember that the answer is given in the question

3 may be seen as \((\sqrt{3})^2\) for M1 only

Condone missing brackets if multiplications are correct