A-level · Sequences & series

Geometric sequences and seriesExam-style questions

Sequences with a constant common ratio — the nth term, the sum of n terms, and the sum to infinity (and exactly when it exists).

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29 questions · A-level level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 2 Level 3 Level 4 Level 5

1
In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

A researcher studies the numbers of students enrolled at two universities, $A$ $A$ and $B$ $B$ . In this question, $t$ $t$ is the time in years since the start of the study.

At the start of the study, $14\,000$ $14000$ students were enrolled at university $A$ $A$ . The researcher models the number of students enrolled at university $A$ $A$ as increasing by $3.5\%$ $3.5%$ each year.
(a) Find, according to this model, the number of students enrolled at university $A$ $A$ when $t = 8$ $t = 8$ . Give your answer to 3 significant figures.

(2 marks)

For university $B$ $B$ , the researcher has the following data.
- When $t = 1$ , $34\,300$ students were enrolled at university $B$
- When $t = 6$ , $31\,000$ students were enrolled at university $B$
The researcher models the number of students enrolled at university $B$ $B$ , $N$ $N$ , by the equation $N = ab^{t}$ $N = a b^{t}$ , where $a$ $a$ and $b$ $b$ are constants.
(b) (i) Show that $b = 0.98$ $b = 0.98$ to 2 significant figures.

(ii) Find the value of $a$ $a$ , giving your answer to 2 significant figures.

(3 marks)

According to the two models, the two universities have the same number of students enrolled when $t = T$ $t = T$
(c) Using $b = 0.98$ $b = 0.98$ and your value of $a$ $a$ , find the value of $T$ $T$ . Give your answer to one decimal place.

(3 marks)
(d) Find, according to the model for university $B$ $B$ , the smallest integer value of $t$ $t$ for which fewer than $25\,000$ $25000$ students are enrolled at university $B$ $B$

(3 marks)
Level 5 · 11 marksTry it
2
In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

A geometric series has first term $16\cos\theta$ , second term $9 + 6\sin\theta$ and third term $18\tan\theta$ , where $\theta$ is a constant.
(a) Show that

$4\sin^2\theta - 20\sin\theta + 9 = 0$

(3 marks)

Given that $\dfrac{\pi}{2} < \theta < \pi$ ,
(b) hence find the exact value of $\theta$ .

(2 marks)
(c) Find the sum to infinity of the series, giving your answer in the form $a + b\sqrt{3}$ , where $a$ and $b$ are integers.

(5 marks)
Level 5 · 10 marksTry it