# Sequences

> Edexcel International GCSE Mathematics A · 4MA1 2016
> Source: https://www.owlsprep.com/study/edexcel-igcse-math-a-s3-sequences/

This guide covers all sequence content for Edexcel IGCSE Maths A (4MA1), including term-to-term/position-to-term rules, linear nth term expressions for arithmetic sequences, and Higher Tier content for finding a, d and series sums.

**Prerequisites:** [Basic arithmetic operations with positive and negative integers](https://www.owlsprep.com/study/edexcel-igcse-math-a-s1-arithmetic/); [Substitution into algebraic expressions](https://www.owlsprep.com/study/edexcel-igcse-math-a-s2-algebra-basics/)

## Learning objectives

- Generate sequence terms using term-to-term and position-to-term rules for common sequence types
- Derive linear nth term expressions for arithmetic sequences (Foundation & Higher)
- Calculate first term (a) and common difference (d) from given arithmetic sequence terms (Higher only)
- Recall and use the nth term formula $a + (n-1)d$ for arithmetic sequences (Higher only)
- Compute the sum of the first n terms of an arithmetic series using the provided formula (Higher only)

## Term-to-Term and Position-to-Term Sequence Rules

Sequences are ordered lists of numbers (called terms) that follow a consistent rule. The two most common ways to define sequences are term-to-term rules and position-to-term rules. Term-to-term rules describe how to move from one term to the next, while position-to-term rules calculate any term using its position n in the sequence.

**Common Sequence Types** — Standard sequences you will encounter include odd numbers, even numbers, square numbers, multiples of a number, and powers (e.g. powers of 2).

**Worked example:** A sequence follows the term-to-term rule: multiply by 2 and subtract 3. The first term is 4. Write the first 4 terms of the sequence.

1. 1st term (n=1): given as 4
2. 2nd term: $(4 \times 2) - 3 = 8 - 3 = 5$
3. 3rd term: $(5 \times 2) - 3 = 10 - 3 = 7$
4. 4th term: $(7 \times 2) - 3 = 14 - 3 = 11$

**Check your understanding**

1. A sequence has position-to-term rule $3n + 2$. What is the 5th term?

   - 15
   - 17
   - 19
   - 21

   *Why:* Substitute n=5: $3(5)+2 = 15+2=17$.

> **Exam tip:** Always check 2-3 terms after deriving a rule to confirm it matches the given sequence.

*Calculator:* allowed

## Linear nth Term of Arithmetic Sequences (Foundation & Higher)

Arithmetic sequences are sequences where the difference between consecutive terms is constant (called the common difference). For Foundation Tier, you need to write the linear nth term expression for these sequences, which takes the form $dn + c$, where d is the common difference and c is a constant.

**Linear nth Term Rule (Foundation)** — To find the nth term: 1. Calculate the common difference d between consecutive terms. 2. Write $dn$, then substitute n=1 to find the constant c needed to match the first term.

**Worked example:** Find the nth term of the sequence 5, 9, 13, 17, ...

1. Calculate the common difference d: $9 - 5 = 4$, $13 - 9 = 4$, so $d=4$.
2. Start with the expression $4n$. Substitute n=1: $4(1) = 4$.
3. The first term of the sequence is 5, so we need to add 1 to 4 to get 5: $c=1$.
4. Final nth term: $4n + 1$

> **Exam tip:** If the sequence is decreasing, your common difference d will be negative, e.g. for 10, 7, 4, 1..., $d = -3$.

*Calculator:* allowed

## Arithmetic Sequences: a and d Formulation (Higher Only)

For Higher Tier, you will use the formal notation for arithmetic sequences: $a$ = first term, $d$ = common difference. The nth term formula is $u_n = a + (n-1)d$, which you must memorise as it is not provided on the formula sheet. You will often be asked to find a and d given two terms of the sequence.

**nth Term Formula (Higher Only)** — Calculates the value of the nth term of an arithmetic sequence, where $a$ = first term, $d$ = common difference, $n$ = position of term.

*Notation:* $u_n = a + (n-1)d$

**Worked example:** The 2nd term of an arithmetic sequence is 7, and the 5th term is 19. Find the values of a and d.

1. Write equations for the given terms using the nth term formula:
2. $$u_2 = a + (2-1)d = a + d = 7$$
3. $$u_5 = a + (5-1)d = a + 4d = 19$$
4. Subtract the first equation from the second to eliminate a: $(a + 4d) - (a + d) = 19 - 7$ → $3d = 12$ → $d = 4$
5. Substitute d=4 back into the first equation: $a + 4 =7$ → $a = 3$
6. Check with the 5th term: $3 + 4(4) = 19$, which matches the given value.

> **warning**
>
> Remember the (n-1) term! A common mistake is writing $u_n = a + nd$, which will give you the wrong value for all terms.

*Calculator:* allowed

## Sum of an Arithmetic Series (Higher Only)

An arithmetic series is the sum of the terms of an arithmetic sequence. The formula for the sum of the first n terms $S_n$ is provided on the Higher Tier formula sheet: $S_n = \frac{n}{2}[2a + (n-1)d]$. You can use this formula directly once you know a, d and n.

**Worked example:** Find the sum of the first 50 terms of the arithmetic series: 4 + 7 + 10 + 13 + ...

1. Identify the required values: first term $a = 4$, common difference $d = 7-4 = 3$, number of terms $n=50$.
2. Substitute into the sum formula:
3. $$S_{50} = \frac{50}{2}[2(4) + (50-1)(3)]$$
4. $$= 25[8 + 49(3)] = 25[8 + 147] = 25[155]$$
5. $$= 3875$$

> **Exam tip:** If you are asked for the sum of terms between two positions (e.g. sum from term 10 to term 20), calculate $S_{20} - S_9$ to get the correct total.

*Calculator:* allowed

## Common pitfalls

- **Wrong:** Using $a + nd$ instead of $a + (n-1)d$ for the nth term (Higher Tier)
  - Why it fails: The first term is n=1, so the difference is added (n-1) times, not n times, leading to off-by-one errors.
  - Correct: Memorise the formula $u_n = a + (n-1)d$ and test it against the first term to confirm it gives the right value when n=1.
- **Wrong:** Confusing term-to-term and position-to-term rules
  - Why it fails: If a question asks for the nth term, a term-to-term rule (e.g. add 4) will get zero marks, as you need to give a formula in terms of n.
  - Correct: Check the question wording carefully: if it asks for the nth term or position-to-term rule, give an expression with n, not a recurrence rule.
- **Wrong:** Using a positive d for decreasing sequences
  - Why it fails: If the sequence is getting smaller each time, the common difference is negative, so your nth term will be incorrect if you use a positive d.
  - Correct: Always calculate d as next term minus previous term, even if it gives a negative value, and test your nth term against the second term to confirm.
- **Wrong:** Calculating $S_{20} - S_{10}$ for the sum of terms from 10 to 20
  - Why it fails: Subtracting S_10 removes the 10th term, so you miss it from your total sum.
  - Correct: Subtract $S_{n-1}$ from $S_m$ when calculating the sum between term n and term m, e.g. $S_{20} - S_9$ for terms 10 to 20.
- **Wrong:** Trying to use geometric sequence rules for non-arithmetic sequences
  - Why it fails: Geometric sequences are out of scope for Edexcel IGCSE Maths A (4MA1), so all sequence questions will be arithmetic or use simple term-to-term rules.
  - Correct: Only use arithmetic sequence formulae for sequences with constant difference; for other sequences, use the given term-to-term rule directly.

## Cheatsheet

| Concept | Foundation Tier | Higher Tier |
| --- | --- | --- |
| Term-to-term rule | Describes how to get next term from previous (e.g. add 4) | Same as Foundation |
| Position-to-term rule | Linear expression for nth term: $dn + c$ | Memorised formula: $u_n = a + (n-1)d$ |
| Find a and d | Not required | Solve simultaneous equations from given terms |
| Sum of first n terms | Not required | Use provided formula: $S_n = \frac{n}{2}[2a + (n-1)d]$ |

## What's next

Now that you have mastered sequences for Edexcel IGCSE Maths A (4MA1), you are ready to progress to related topics in the Sequences, Functions and Graphs unit. Next, you will learn how to plot and interpret linear graphs, which builds directly on your understanding of linear nth term expressions. For Higher Tier students, this sequence knowledge will also be used to solve problems involving algebraic patterns and real-world applications that follow arithmetic rules. Make sure to practice past paper questions for sequences to familiarise yourself with common exam phrasing and avoid the common pitfalls outlined in this guide, as this topic appears frequently in both Foundation and Higher papers.

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