Cambridge A Level Mathematics 9709 · Pure 1

Arithmetic and geometric progressions

An arithmetic progression (AP) adds the same number each time; a geometric progression (GP) multiplies by the same number each time. For an AP with first term $a$ and common difference $d$ , the nth term is $a + (n - 1) d$ ; for a GP with common ratio $r$ it is $a r^{n - 1}$ . A GP has a sum to infinity, $\frac{a}{1 - r}$ , only when $∣ r ∣ < 1$ .

The formulas are in the MF19 booklet, so the marks on Cambridge 9709 Paper 1 are in setting up the equations — especially when one question links an AP and a GP.

Updated 28 September 2026

How much it is worth

Arithmetic and geometric progressions: 335 marks across 42 real Cambridge 9709 papers — 10.6% of the total, and it came up on every one of them. How that ranks against every other 9709 topic →

Arithmetic progressions

u_{n} = a + (n - 1) d S_{n} = \frac{n}{2} {2 a + (n - 1) d} = \frac{n}{2} (a + l)

$l$ is the last term. Use the second sum formula when you know the first and last terms — it saves finding $d$ .

Most AP questions give two facts and ask for $a$ and $d$ : “the 5th term is 17 and the sum of the first 10 terms is 185” becomes two simultaneous equations, $a + 4 d = 17$ and $5 (2 a + 9 d) = 185$ , giving $a = 5, d = 3$ .

A sum from the $k$ th term to the $m$ th term is $S_{m} - S_{k - 1}$ — not $S_{m} - S_{k}$ , which drops the $k$ th term itself.

Geometric progressions

u_{n} = a r^{n - 1} S_{n} = \frac{a (1 - r^{n})}{1 - r} (r \neq 1)

The common ratio is any term divided by the one before it: $r = \frac{u_{2}}{u_{1}} = \frac{u_{3}}{u_{2}}$ . Three terms $x, y, z$ are consecutive terms of a GP exactly when $y^{2} = x z$ — the condition behind most linked questions.

Growth and decay are GPs in disguise. “Increases by 5% each year” means $r = 1.05$ ; “loses 20% of its value each year” means $r = 0.8$ .

Sum to infinity

S_{\infty} = \frac{a}{1 - r} only when ∣ r ∣ < 1, i.e. - 1 < r < 1

When $∣ r ∣ < 1$ , $r^{n} \to 0$ as $n$ grows, so $S_{n}$ settles towards $\frac{a}{1 - r}$ . A question that asks you to “show that the series is convergent” or “explain why the sum to infinity exists” wants that condition stated with the value of $r$ .

When $r$ is given in terms of $x$ — say $r = 2 \cos x$ or $r = \frac{x}{3}$ — the convergence condition becomes an inequality to solve for $x$ , which is how this topic meets trigonometry and quadratic inequalities.

An AP and a GP in the same question

The highest-value questions tie the two together: “the first, third and ninth terms of an AP are the first three terms of a GP”. The method never changes:

Write each named AP term in terms of $a$ and $d$ .
Use the GP condition — equal ratios, or $y^{2} = x z$ — to get one equation.
Solve, and reject any root the question rules out (usually $d = 0$ or $r = 1$ ).
Only then find $r$ and whatever sum is asked for.

Worked examples

Worked example

The first term of an arithmetic progression is 2 and the common difference is d, where d ≠ 0. The first, third and ninth terms of the AP are the first three terms of a geometric progression. Find d, the common ratio of the GP, and the sum of the first 20 terms of the AP.

AP terms: $u_{1} = 2$ , $u_{3} = 2 + 2 d$ , $u_{9} = 2 + 8 d$ .

GP condition, middle term squared equals the product of the other two:

(2 + 2 d)^{2} = 2 (2 + 8 d) \Rightarrow 4 + 8 d + 4 d^{2} = 4 + 16 d \Rightarrow 4 d^{2} - 8 d = 0

So $4 d (d - 2) = 0$ , and since $d \neq 0$ , $d = 2$ . The GP is 2, 6, 18, so $r = 3$ .

S_{20} = \frac{20}{2} {2 (2) + 19 (2)} = 10 \times 42 = 420

Worked example

A geometric progression has first term 12 and sum to infinity 48. Find the common ratio and the sum of the first five terms.

\frac{12}{1 - r} = 48 \Rightarrow 1 - r = \frac{1}{4} \Rightarrow r = 0.75

$∣ r ∣ < 1$ , so the sum to infinity exists, as the question assumes.

S_{5} = \frac{12 (1 - {0.75}^{5})}{1 - 0.75} = 48 (1 - 0.2373 \dots) = 36.6 (3 s.f.)

Common mistakes

1.Using n instead of n − 1 in the nth term.
The first term has no $d$ added (or no $r$ multiplied), so the nth term has $n - 1$ of them.
2.Using the sum to infinity when |r| ≥ 1.
Check $- 1 < r < 1$ first. With $r = 2$ the terms grow without limit and $\frac{a}{1 - r}$ gives a meaningless negative number.
3.Subtracting the wrong partial sum.
Terms $k$ to $m$ inclusive: $S_{m} - S_{k - 1}$ .
4.Writing a 5% increase as r = 0.05.
The multiplier is $1.05$ . A 5% decrease is $0.95$ .
5.Keeping the root the question excluded.
“ $d \neq 0$ ” or “the terms are distinct” is there to tell you which solution of the quadratic to throw away — say so in your answer.

Common questions

Are the AP and GP formulas given in 9709?

Yes. The MF19 booklet gives the nth term and sum for both, and the GP sum to infinity with its $∣ r ∣ < 1$ condition. What MF19 does and does not give you →

How do I show three terms form a geometric progression?

Show the ratios are equal: $\frac{y}{x} = \frac{z}{y}$ , which is the same as $y^{2} = x z$ . For an arithmetic progression, show the differences are equal: $y - x = z - y$ .

How much of 9709 Paper 1 is progressions?

335 marks across the 42 Paper 1s Quanta has mapped — 10.6% — on 42 of them. The binomial expansion, the other half of section 1.6, carries another 205.

Arithmetic and geometric progressions on real papers

The papers that leaned on it hardest, as a share of the paper. Each one has its own page on Quanta with the full topic breakdown:

9709/13/M/J/2415 of 75 marks 9709/12/F/M/2513 of 75 marks 9709/11/O/N/2513 of 75 marks

Practise arithmetic and geometric progressions against real mark schemes

Quanta has real Cambridge A Level Maths 9709 past-paper questions, broken into skill checkpoints, marked criterion by criterion the way examiners mark — and it tracks which skills you’re missing. Free for individual students.

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