Geometric sequences with surds and other patterns

3 exam-style questions. Answers and the mark for each step are on the last page.

Question 1Non-calculator · 3 marks
Here are the first four terms of a geometric sequence: $\sqrt{2}, \ 2, \ 2\sqrt{2}, \ 4$ .
(a) Write down the common ratio. Give your answer in exact form. (1)
(b) Work out the 8th term. (2)
Question 2Non-calculator · 4 marks
A geometric sequence has second term 8 and fifth term 64. Its common ratio is positive.
(a) Find the common ratio. (2)
(b) Find the first term. (2)
Question 3Non-calculator · 6 marks
A geometric sequence has first term $2 + \sqrt{3}$ and second term $1$ .
(a) Show that the common ratio is $2 - \sqrt{3}$ . (3)
(b) Find the third term. Give your answer in the form $a + b\sqrt{3}$ . (1)
(c) Find the fourth term. Give your answer in the form $a + b\sqrt{3}$ . (2)

Answers and marks

(a) $\sqrt{2}$

(b) $16$

M1 Writing $(\sqrt{2})^8$ or continuing the sequence: $4\sqrt{2}, 8, 8\sqrt{2}, 16$ .
A1 The correct answer, $16$ .

(a) $2$

(b) $4$

(a) $\frac{1}{2 + \sqrt{3}} = \frac{2 - \sqrt{3}}{4 - 3} = 2 - \sqrt{3}$

(b) $2 - \sqrt{3}$

(c) $7 - 4\sqrt{3}$

Geometric sequences with surds and other patterns

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