Edexcel IAL P2 Sequences and Series: exam questions and answers, notes and worked examples

Q: Are the series formulas in the IAL formula booklet?

On our IAL formula sheet the nth term and sum formulas for arithmetic and geometric series, and the sum to infinity, are all marked In the formula booklet. What it will not tell you is which one a question needs.

Q: What is a periodic sequence?

One that repeats after a fixed number of terms, its order. For example 2, −1, ½, 2, −1, ½, … has order 3. To find the sum of many terms, add one full cycle and multiply by the number of cycles.

Q: Do I have to state |r| < 1 when I find r from a sum to infinity?

Yes: show the check. If a geometric series has first term 3 and sum to infinity 8, then 3/(1 − r) = 8, so r = 5/8. Then write that |r| = 5/8 < 1, so the sum to infinity exists. The value of r on its own is not a complete answer.

Q: Is a calculator allowed in IAL P2?

Yes. A calculator is allowed in every Edexcel IAL maths unit, P2 included. Method marks are for the working you write down, so show it.

Q: How long is the IAL P2 exam?

Each IAL maths unit, P2 included, is one written paper of 1 hour 30 minutes worth 75 marks.

Q: How often does sequences and series come up in P2?

In our mapping of the 4 Pearson P2 papers from June 2024 to June 2025, sequences and series was tested in 4 of them (7 questions in all). Any topic can be examined in any series.

Updated October 2026 · By Pete Bromfield

P2 sequences and series covers sequences defined by an $n$ th term or a recurrence, arithmetic and geometric series, the sum to infinity of a geometric series with $| r | < 1$ , and $Σ$ notation. Decide first whether the question wants one term or a sum, and whether the pattern adds a fixed amount (arithmetic) or multiplies by one (geometric).

P2 · Pure Mathematics 2
Calculator allowed
3 worked examples
35 practice questions
Tested in 4 of the last 4 papers

Practise P2 sequences and series →

Notes
Formulas
Worked examples
Mistakes
FAQ

P2 Sequences and Series notes

Recurrence relations: increasing, decreasing and periodic

A recurrence gives each term from the one before, $u_{n + 1} = f (u_{n})$ , plus a first term. A sequence is increasing if $u_{n + 1} > u_{n}$ for all $n$ , decreasing if $u_{n + 1} < u_{n}$ , and periodic if it repeats. With $u_{1} = 2$ and $u_{n + 1} = \frac{1}{1 - u_{n}}$ : $u_{2} = - 1$ , $u_{3} = \frac{1}{2}$ , $u_{4} = 2$ , so it is periodic with order 3.

Arithmetic sequences and series

$u_{n} = a + (n - 1) d$ and $S_{n} = \frac{n}{2} (2 a + (n - 1) d) = \frac{n}{2} (a + l)$ . With $a = 7$ and $d = 4$ : $u_{15} = 63$ and $S_{15} = 525$ . Two unknowns need two facts, so write each as an equation in $a$ and $d$ .

Geometric sequences and series

$u_{n} = a r^{n - 1}$ and $S_{n} = \frac{a (1 - r^{n})}{1 - r}$ for $r \neq 1$ . With $a = 5$ and $r = 3$ : $S_{6} = 1820$ . Find $r$ by dividing a term by the one before it; when $n$ is the unknown, take logarithms.

Sum to infinity

Only when $| r | < 1$ does a geometric series have a sum to infinity: $S_{\infty} = \frac{a}{1 - r}$ . For $40 - 20 + 10 - \dots$ , $r = - \frac{1}{2}$ and $S_{\infty} = \frac{80}{3}$ .

Sigma notation

$\sum_{r = 1}^{n}$ means add the terms for $r = 1, 2, \dots, n$ . $\sum_{r = 1}^{20} (3 r + 1)$ is arithmetic with first term 4 and last term 61, so it equals $650$ . Count terms carefully: $\sum_{r = 5}^{12}$ has $8$ terms (top minus bottom plus one).

Key formulas

$n$ th terms In the formula booklet: $u_{n} = a + (n - 1) d, u_{n} = a r^{n - 1}$
Arithmetic series In the formula booklet: $S_{n} = \frac{1}{2} n (a + l) = \frac{1}{2} n [2 a + (n - 1) d]$
Geometric series In the formula booklet: $S_{n} = \frac{a (1 - r^{n})}{1 - r}, S_{\infty} = \frac{a}{1 - r} (| r | < 1)$

In the formula booklet = printed in Pearson's IAL formulae booklet, so know where to find it. Learn it = not printed there, so learn it by heart. All P2 formulas on one page (free PDF) →

P2 Sequences and Series worked examples

Each mark scheme is checked line by line before it is published here. M marks are for method, A marks for accurate answers that follow from them, B marks for independent results.

Worked example 1 · 5 marks · hard

The second term of a geometric series is 12 and its sum to infinity is 64. Find the two possible values of the common ratio, $r$ , and the corresponding values of the first term, $a$ .

Mark scheme

M1 for setting up the two equations: $a r = 12$ and $\frac{a}{1 - r} = 64$ .
M1 for substituting $a = \frac{12}{r}$ to get $\frac{12 / r}{1 - r} = 64 ⟹ 12 = 64 r (1 - r)$ .
M1 for forming the quadratic $16 r^{2} - 16 r + 3 = 0$ .
A1 for finding $r = 0.25$ and $r = 0.75$ .
A1 for finding the corresponding first terms: $a = 48$ and $a = 16$ .

5 steps · 1 to fill in

Worked example 2 · 5 marks · hard

The first, third, and seventh terms of an arithmetic sequence (with a non-zero common difference) form three consecutive terms of a geometric sequence. Show that the first term is twice the common difference ( $a = 2 d$ ).

Mark scheme

M1 for writing the 1st, 3rd, and 7th terms as $a, a + 2 d, a + 6 d$ .
M1 for setting up the geometric ratio condition: $\frac{a + 2 d}{a} = \frac{a + 6 d}{a + 2 d}$ .
M1 for cross-multiplying and expanding: $(a + 2 d)^{2} = a (a + 6 d) ⟹ a^{2} + 4 a d + 4 d^{2} = a^{2} + 6 a d$ .
M1 for rearranging to $4 d^{2} - 2 a d = 0 ⟹ 2 d (2 d - a) = 0$ .
A1 for concluding that since $d \neq 0$ , $2 d - a = 0 ⟹ a = 2 d$ .

5 steps

Worked example 3 · 4 marks · medium

Find the least number of terms required for the sum of the arithmetic series $5 + 9 + 13 + 17 + . . .$ to exceed 1000.

Mark scheme

M1 for setting up the inequality: $S_{n} = \frac{n}{2} (2 (5) + (n - 1) 4) > 1000$ .
M1 for simplifying to a quadratic inequality: $n (10 + 4 n - 4) > 2000 ⟹ 4 n^{2} + 6 n - 2000 > 0$ .
M1 for solving the quadratic equation $2 n^{2} + 3 n - 1000 = 0$ to find the critical value $n \approx 21.6$ .
A1 for concluding that since $n$ must be an integer, the least number of terms is $n = 22$ .

4 steps · 1 to fill in

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Common P2 sequences and series mistakes

n instead of n − 1. The $n$ th term is $a + (n - 1) d$ or $a r^{n - 1}$ : the first term has no $d$ added and no $r$ multiplied.
A sum to infinity that does not exist. $S_{\infty}$ needs $| r | < 1$ . If $r$ involves $x$ , that condition gives the range of $x$ for which the sum exists.
Miscounting sigma terms. From $r = 5$ to $r = 12$ is 8 terms, not 7.
Rounding n the wrong way. For "the least number of terms so that the sum exceeds 1000", solve, round up, and check the sums either side.

Every P2 mistake from the examiner reports, with the paper it came from →

On your calculator: how to find terms and sums of sequences. A calculator is allowed in every IAL unit.

P2 Sequences and Series FAQ

Are the series formulas in the IAL formula booklet?

On our IAL formula sheet the nth term and sum formulas for arithmetic and geometric series, and the sum to infinity, are all marked In the formula booklet. What it will not tell you is which one a question needs.

What is a periodic sequence?

One that repeats after a fixed number of terms, its order. For example 2, −1, ½, 2, −1, ½, … has order 3. To find the sum of many terms, add one full cycle and multiply by the number of cycles.

Do I have to state |r| < 1 when I find r from a sum to infinity?

Yes: show the check. If a geometric series has first term 3 and sum to infinity 8, then 3/(1 − r) = 8, so r = 5/8. Then write that |r| = 5/8 < 1, so the sum to infinity exists. The value of r on its own is not a complete answer.

Is a calculator allowed in IAL P2?

Yes. A calculator is allowed in every Edexcel IAL maths unit, P2 included. Method marks are for the working you write down, so show it.

How long is the IAL P2 exam?

Each IAL maths unit, P2 included, is one written paper of 1 hour 30 minutes worth 75 marks.

How often does sequences and series come up in P2?

In our mapping of the 4 Pearson P2 papers from June 2024 to June 2025, sequences and series was tested in 4 of them (7 questions in all). Any topic can be examined in any series.

Next steps

More P2: Logarithms and Exponentials · Circles · Further Trigonometry · Algebraic Methods · Proof · The Binomial Expansion · Differentiation (stationary points) · Integration (definite integrals and area)

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Other units: IAL P1 · IAL P3 · IAL P4 · IAL S1 · IAL M1

Edexcel IAL P2 Sequences and Series: exam questions and answers, notes and worked examples

Try it first

P2 Sequences and Series notes

Recurrence relations: increasing, decreasing and periodic

Arithmetic sequences and series

Geometric sequences and series

Sum to infinity

Sigma notation

Key formulas

P2 Sequences and Series worked examples

Worked example 1 · 5 marks · hard

Mark scheme

Worked example 2 · 5 marks · hard

Mark scheme

Worked example 3 · 4 marks · medium

Mark scheme

Common P2 sequences and series mistakes

P2 Sequences and Series FAQ

Are the series formulas in the IAL formula booklet?

What is a periodic sequence?

Do I have to state |r| < 1 when I find r from a sum to infinity?

Is a calculator allowed in IAL P2?

How long is the IAL P2 exam?

How often does sequences and series come up in P2?

Next steps

Your P2 revision kit