A-level · Sequences & series

Arithmetic sequences and seriesExam-style questions

Sequences with a constant common difference — the nth-term rule and the sum formula, and where each comes from.

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34 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
Anna saves money each week. She saves $\pounds 15$ in the first week, and increases the amount by $\pounds 4$ each week.
(a) Find how much she saves in week $20$ . (2 marks)
(b) Find the total she has saved after $20$ weeks. (2 marks)
(c) Find the smallest number of weeks for her total savings to exceed $\pounds 2000$ . (4 marks)
Level 5 · 8 marksTry it
2
An arithmetic sequence has third term $13$ and eighth term $33$ .
(a) Find the first term $a$ and common difference $d$ . (3 marks)
(b) Show that the sum of the first $n$ terms is $S_n = 2n^2 + 3n$ . (3 marks)
Level 5 · 6 marksTry it
3
The sum of the first $n$ terms of a sequence is $S_n = 2n^2 + 3n$ .
(a) Find an expression for the $n$ th term $u_n$ . (4 marks)
(b) Hence state, with a reason, whether the sequence is arithmetic. (2 marks)
Level 5 · 6 marksTry it
4
(a) The first term of an arithmetic series is $a$ , and $d$ is its common difference.

Prove that

$S_n = \frac{n}{2}\left[2a + (n - 1)d\right]$

where $S_n$ is the sum of the first $n$ terms of the series.

(3 marks)

Dev borrows £ $3000$ from a relative. He repays the loan in monthly instalments.

His repayment in month $1$ is £ $55$ , and each month after that he repays £ $10$ more than he did the month before.
(b) Show that Dev's repayment in month $10$ is £ $145$

(1 mark)

Given that Dev takes exactly $n$ months to repay the loan completely,
(c) show that

$n^2 + 10n - 600 = 0$

(2 marks)
(d) Solve the equation in part (c).

(2 marks)
(e) Hence state how many months it takes Dev to repay the loan, giving a reason for your answer.

(1 mark)
Level 5 · 9 marksTry it
5
A publisher launches a new weekly magazine. In Week 1, $1500$ $1500$ copies of the magazine are sold. The publisher considers two models for the number of copies sold in later weeks.
- Model A: the number of copies sold increases by $300$ each week.
- Model B: the number of copies sold increases by $9\%$ each week.
(a) Verify that the two models' predictions for the number of copies sold in Week 10 differ by $942$ $942$ , correct to the nearest whole number.

(3 marks)

Week $X$ $X$ is the first week for which model B predicts more copies sold than model A.
(b) Show that $X$ $X$ satisfies the inequality $X > 1 + \log_{1.09}(0.2X + 0.8)$ $X > 1 + lo g_{1.09} (0.2 X + 0.8)$ .

(3 marks)
(c) Use the iterative formula $x_{n+1} = 1 + \log_{1.09}(0.2x_n + 0.8)$ $x_{n + 1} = 1 + lo g_{1.09} (0.2 x_{n} + 0.8)$ with $x_1 = 15$ $x_{1} = 15$ to find the value of $X$ $X$ .

You should show the result of each iteration.

(3 marks)
(d) (i) Suggest a reason why model A may give unreliable predictions for later weeks.

(ii) Suggest a different reason why model B may give unreliable predictions for later weeks.

(2 marks)
Level 5 · 11 marksTry it