Arithmetic and Geometric Series

Q: Do I need to memorize the AP/GP sum formulae?

No, all sum formulae for AP and GP (including sum to infinity) are provided on the official 4PM1 formula sheet. You only need to memorize the nth term formulae for AP and GP.

Q: When can I use the sum to infinity formula?

You can only use the sum to infinity formula for geometric series where the absolute value of the common ratio is less than 1, i.e. . If , the series diverges and has no finite sum to infinity.

Q: Do I need to prove the sum formulae for AP/GP in the exam?

No, proofs of all AP and GP sum formulae are explicitly excluded from the 4PM1 specification, so you will not be asked to derive them.

Edexcel International GCSE Further Pure Mathematics· S5 5B· 25 min read

1. Arithmetic Series (AP) Fundamentals★★☆☆☆⏱ 6 min

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📘 Definition

Arithmetic Progression (AP)

A sequence where each term increases or decreases by a fixed constant value called the common difference, $d$ .

Example:

The sequence 2, 5, 8, 11, 14 is an AP with first term $a = 2$ and common difference $d = 3$ .

The nth term of an AP is calculated with the formula $u_{n} = a + (n - 1) d$ , which you must memorize for the exam. The sum of the first $n$ terms of an AP is given on your formula sheet as $S_{n} = \frac{n}{2} [2 a + (n - 1) d]$ , so you do not need to memorize this.

📐 Worked Example

An AP has first term 7 and common difference 3. Find the 12th term and the sum of the first 20 terms.

1
Identify known values: $a = 7$ , $d = 3$ , $n = 12$ for the nth term calculation.
2
$u_{12} = 7 + (12-1) \times 3 = 7 + 33 = 40$
3
For the sum of the first 20 terms, use $n = 20$ in the sum formula:
4
$S_{20} = \frac{20}{2}[2(7) + (20-1) \times 3] = 10[14 + 57] = 10 \times 71 = 710$

Exam tip:

Always label your values clearly (a, d, n) at the start of AP problems to avoid substitution errors.

2. Geometric Series (GP) Fundamentals★★★☆☆⏱ 7 min

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📘 Definition

Geometric Progression (GP)

A sequence where each term is multiplied by a fixed constant value called the common ratio, $r$ , to get the next term.

Example:

The sequence 3, 6, 12, 24, 48 is a GP with first term $a = 3$ and common ratio $r = 2$ .

The nth term of a GP is calculated with the formula $u_{n} = a r^{n - 1}$ , which you must memorize for the exam. The sum of the first $n$ terms of a GP is given on your formula sheet as $S_{n} = \frac{a ( 1 - r ^{n} )}{1 - r}$ ; use this form regardless of whether $r$ is greater than or less than 1 to avoid sign errors.

📐 Worked Example

A GP has first term 128 and common ratio 0.5. Find the 6th term and the sum of the first 8 terms.

1
Identify known values: $a = 128$ , $r = 0.5$ , $n = 6$ for the nth term calculation.
2
$u_6 = 128 \times (0.5)^{6-1} = 128 \times (0.5)^5 = 128 \times \frac{1}{32} = 4$
3
For the sum of the first 8 terms, use $n = 8$ in the sum formula:
4
$S_8 = \frac{128(1 - (0.5)^8)}{1 - 0.5} = \frac{128(1 - \frac{1}{256})}{0.5} = 256 \times \frac{255}{256} = 255$

Exam tip:

When $r$ is a fraction or negative, use brackets around $r$ when raising it to a power to avoid sign or calculation errors.

3. Convergent Geometric Series and Sum to Infinity★★★☆☆⏱ 6 min

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📘 Definition

Convergent Geometric Series

A geometric series where terms get progressively closer to 0 as $n$ increases, so the total sum of infinitely many terms is a finite value.

Example:

The GP 16, 8, 4, 2, 1, ... has $r = 0.5$ , so it is convergent, with a finite sum to infinity.

The sum to infinity of a convergent GP is given on your formula sheet as $S_{\infty} = \frac{a}{1 - r}$ . You may only use this formula if the absolute value of the common ratio is less than 1, i.e. $∣ r ∣ < 1$ . If $∣ r ∣ \geq 1$ , the series diverges and has no finite sum to infinity.

📐 Worked Example

A GP has first term 50 and sum to infinity 62.5. Find the common ratio $r$ , and confirm the series is convergent.

1
Substitute known values into the sum to infinity formula:
2
$62.5 = \frac{50}{1 - r}$
3
Rearrange to solve for $r$ :
4
$62.5(1 - r) = 50 \implies 1 - r = \frac{50}{62.5} = 0.8 \implies r = 1 - 0.8 = 0.2$
5
Check convergence: $∣0.2∣ = 0.2 < 1$ , so the series is convergent.

Exam tip:

If an exam question asks for the sum to infinity, always state the $∣ r ∣ < 1$ check as part of your answer to get full marks.

4. Mixed AP/GP Exam Problems★★★★☆⏱ 8 min

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Most exam questions for this topic combine AP and GP concepts, so you will need to distinguish between the two sequence types from context, and apply the correct formulae as required.

📐 Worked Example

The first three terms of a sequence are 8, 13, 18. (a) State if this is an AP or GP, and find the 25th term. (b) A second sequence has first term 2 and third term 18, and is a GP. Find the sum of the first 10 terms of this GP.

1
Part (a): Check the difference between consecutive terms: $13 - 8 = 5$ , $18 - 13 = 5$ . This is an AP with $a = 8$ , $d = 5$ .
2
$u_{25} = 8 + (25 - 1) \times 5 = 8 + 120 = 128$
3
Part (b): For the GP, first term $a = 2$ , third term $a r^{2} = 18$ :
4
$2r^2 = 18 \implies r^2 = 9 \implies r = 3 \text{ or } r = -3$
5
Calculate sum for $r = 3$ :
6
$S_{10} = \frac{2(1 - 3^{10})}{1 - 3} = \frac{2(1 - 59049)}{-2} = 59048$
7
Calculate sum for $r = - 3$ :
8
$S_{10} = \frac{2(1 - (-3)^{10})}{1 - (-3)} = \frac{2(1 - 59049)}{4} = -29524$

Exam tip:

When solving for $r$ in a GP, remember that $r$ can be negative unless the question specifies all terms are positive, so you may have two valid solutions.

5. Common Pitfalls

Wrong move:

Using the AP nth term formula for a GP, or vice versa.

Why:

Confusing the linear AP nth term with the exponential GP nth term leads to completely incorrect answers.

Correct move:

First check if the sequence has a constant difference (AP) or constant ratio (GP) before selecting a formula.

Wrong move:

Using the sum to infinity formula for a GP with $∣ r ∣ \geq 1$ .

Why:

The series diverges, so there is no finite sum, and this will give an invalid answer.

Correct move:

Always check that $∣ r ∣ < 1$ before applying the sum to infinity formula, and state this check in your answer.

Wrong move:

Using $n$ instead of $n - 1$ in the nth term formula for AP/GP.

Why:

The first term is when $n = 1$ , so the number of increments/multipliers is always one less than $n$ .

Correct move:

Substitute $n = 1$ into your nth term calculation to verify it gives the correct first term $a$ .

Wrong move:

Forgetting that $r$ can be negative in GP problems.

Why:

Many questions omit that terms are positive, so negative $r$ is a valid solution that you will lose marks for missing.

Correct move:

When solving for $r$ from an even power (e.g. $r^{2} = 16$ ), always include both positive and negative roots unless specified otherwise.

Wrong move:

Incorrectly rearranging the sum to infinity formula to solve for $r$ or $a$ .

Why:

Algebraic errors when rearranging fractions are common under exam pressure.

Correct move:

Cross-multiply first to eliminate the denominator before rearranging terms, then substitute back to check your answer.

6. Quick Reference Cheatsheet

Concept	Formula	Given on Formula Sheet?	Key Condition
AP nth term	$u_{n} = a + (n - 1) d$	No	$d$ = constant common difference
AP sum to $n$ terms	$S_{n} = \frac{n}{2} [2 a + (n - 1) d]$	Yes
GP nth term	$u_{n} = a r^{n - 1}$	No	$r$ = constant common ratio
GP sum to $n$ terms	$S_{n} = \frac{a ( 1 - r ^{n} )}{1 - r}$	Yes
GP sum to infinity	$S_{\infty} = \frac{a}{1 - r}$	Yes	$∣ r ∣ < 1$ (convergent series only)

7. Frequently Asked

Do I need to memorize the AP/GP sum formulae?

No, all sum formulae for AP and GP (including sum to infinity) are provided on the official 4PM1 formula sheet. You only need to memorize the nth term formulae for AP and GP.

When can I use the sum to infinity formula?

You can only use the sum to infinity formula for geometric series where the absolute value of the common ratio is less than 1, i.e. $∣ r ∣ < 1$ . If $∣ r ∣ \geq 1$ , the series diverges and has no finite sum to infinity.

Do I need to prove the sum formulae for AP/GP in the exam?

No, proofs of all AP and GP sum formulae are explicitly excluded from the 4PM1 specification, so you will not be asked to derive them.

Going deeper

referenceEdexcel 4PM1 Full Formula SheetOfficial formula sheet for the 2016 specification

What's Next

Now that you have mastered arithmetic and geometric series for Edexcel IGCSE Further Pure Mathematics, you are ready to move on to the next topic in the series unit, binomial series. You should also practice past paper questions on AP/GP to reinforce your problem-solving speed and accuracy, as this topic appears on almost every exam paper. Make sure you can quickly distinguish between AP and GP contexts, and always double-check your substitutions into formulae to avoid avoidable errors. Remember that all sum formulae are given on your formula sheet, so focus on memorizing the nth term formulae and the $∣ r ∣ < 1$ condition for sum to infinity.