# Arithmetic and Geometric Series

> Edexcel International GCSE Further Pure Mathematics · 4PM1 2016 Spec
> Source: https://www.owlsprep.com/study/edexcel-igcse-further-pure-math-s5-arithmetic-and-geometric-series/

This guide covers all required arithmetic and geometric series content for Edexcel IGCSE Further Pure Mathematics (4PM1) S5, including nth term calculations, sum to n terms, and sum to infinity for convergent geometric series, with worked exam-style examples.

**Prerequisites:** [Algebraic substitution and rearrangement](https://www.owlsprep.com/study/edexcel-igcse-math-algebra-rearrangement/); [Absolute value basics](https://www.owlsprep.com/study/edexcel-igcse-math-number-absolute-value/)

## Learning objectives

- Recall and apply the nth term formula for arithmetic and geometric sequences
- Calculate the sum of the first n terms of arithmetic and geometric series using given formulae
- Use the sum to infinity formula for convergent geometric series and apply the |r| < 1 condition
- Solve structured exam problems combining arithmetic and geometric series concepts

## Arithmetic Series (AP) Fundamentals

**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 $S_n = \frac{n}{2}[2a + (n-1)d]$, 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.

*Calculator:* allowed

## Geometric Series (GP) Fundamentals

**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 $u_n = ar^{n-1}$, 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.

*Calculator:* allowed

## Convergent Geometric Series and Sum to Infinity

**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.

*Calculator:* allowed

## Mixed AP/GP Exam Problems

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 $ar^2 = 18$:
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.

*Calculator:* allowed

## Common pitfalls

- **Wrong:** Using the AP nth term formula for a GP, or vice versa.
  - Why it fails: Confusing the linear AP nth term with the exponential GP nth term leads to completely incorrect answers.
  - Correct: First check if the sequence has a constant difference (AP) or constant ratio (GP) before selecting a formula.
- **Wrong:** Using the sum to infinity formula for a GP with $|r| \geq 1$.
  - Why it fails: The series diverges, so there is no finite sum, and this will give an invalid answer.
  - Correct: Always check that $|r| < 1$ before applying the sum to infinity formula, and state this check in your answer.
- **Wrong:** Using $n$ instead of $n-1$ in the nth term formula for AP/GP.
  - Why it fails: The first term is when $n=1$, so the number of increments/multipliers is always one less than $n$.
  - Correct: Substitute $n=1$ into your nth term calculation to verify it gives the correct first term $a$.
- **Wrong:** Forgetting that $r$ can be negative in GP problems.
  - Why it fails: Many questions omit that terms are positive, so negative $r$ is a valid solution that you will lose marks for missing.
  - Correct: When solving for $r$ from an even power (e.g. $r^2 = 16$), always include both positive and negative roots unless specified otherwise.
- **Wrong:** Incorrectly rearranging the sum to infinity formula to solve for $r$ or $a$.
  - Why it fails: Algebraic errors when rearranging fractions are common under exam pressure.
  - Correct: Cross-multiply first to eliminate the denominator before rearranging terms, then substitute back to check your answer.

## 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 | $S_n = \frac{n}{2}[2a + (n-1)d]$ | Yes |  |
| GP nth term | $u_n = ar^{n-1}$ | 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) |

## 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.

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