What this chapter is about
This chapter introduces you to the idea of sequences — ordered lists of numbers that follow a definite pattern or rule. You will learn how to observe a pattern, describe it mathematically, and use it to predict future terms. This skill is fundamental because patterns appear everywhere: in nature, in finance, in science experiments, and in everyday life.
The focus here is on understanding what makes a sequence, recognising different types of progressions, and learning to find any term of a sequence when you know its rule. You will study arithmetic progressions in detail, where each term differs from the previous one by a fixed amount. By the end of this chapter, you should be able to identify whether a given list forms a sequence, write the general term of simple sequences, and calculate specific terms without listing every number before them.
This chapter builds on your knowledge of linear equations and algebraic expressions from earlier classes. The reasoning you develop here will prepare you for more advanced work on series and mathematical induction in higher classes.
Key ideas
- A sequence is an ordered list of numbers written in a definite pattern. The position of each number matters: the first term, second term, third term, and so on.
- Each number in a sequence is called a term. We often write the nth term as aₙ, where n is the position (1, 2, 3, …).
- A rule or formula connects the position number n to the value of the term aₙ. For example, if aₙ = 2n + 1, then a₁ = 3, a₂ = 5, a₃ = 7, and so on.
- An arithmetic progression (AP) is a sequence where the difference between any two consecutive terms is always the same. This fixed difference is called the common difference, denoted by d.
- In an AP with first term a and common difference d, the nth term is given by: aₙ = a + (n − 1) × d.
- Not every list of numbers is a sequence. A sequence must have a clear rule that determines each term from its position.
- Recognising whether differences between consecutive terms are constant helps you decide if a list forms an arithmetic progression.
Formulas and facts to remember
- nth term of an AP: aₙ = a + (n − 1) × d, where a is the first term and d is the common difference.
- Common difference: d = a₂ − a₁ = a₃ − a₂ = … (the same throughout the AP).
- General term of a sequence: If a rule like aₙ = 3n − 2 is given, substitute the position n to find any term directly.
- Checking for AP: Calculate differences between consecutive terms. If all differences are equal, the sequence is an AP.
- First term: Always the term when n = 1, so a₁ = a.
Worked examples
Example 1: Finding terms of a sequence from its rule
The nth term of a sequence is given by aₙ = 4n − 3. Find the first four terms.
Solution:
- For n = 1: a₁ = 4 × 1 − 3 = 4 − 3 = 1
- For n = 2: a₂ = 4 × 2 − 3 = 8 − 3 = 5
- For n = 3: a₃ = 4 × 3 − 3 = 12 − 3 = 9
- For n = 4: a₄ = 4 × 4 − 3 = 16 − 3 = 13
The first four terms are 1, 5, 9, 13.
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Example 2: Checking whether a list forms an AP
Is the list 7, 12, 17, 22 an arithmetic progression?
Solution: Calculate the differences between consecutive terms:
- 12 − 7 = 5
- 17 − 12 = 5
- 22 − 17 = 5
Since all differences are equal (d = 5), this list is an AP with first term 7 and common difference 5.
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Example 3: Finding a specific term of an AP
A bus company adds 3 new buses every year. In the first year, they had 15 buses. How many buses will they have in the 10th year?
Solution: This forms an AP with first term a = 15 and common difference d = 3.
Using the formula aₙ = a + (n − 1) × d: a₁₀ = 15 + (10 − 1) × 3 a₁₀ = 15 + 9 × 3 a₁₀ = 15 + 27 a₁₀ = 42
The company will have 42 buses in the 10th year.
Common mistakes
- Writing the nth term formula as aₙ = a + n × d instead of aₙ = a + (n − 1) × d → Remember that the first term has n = 1, so we multiply d by (n − 1), not n.
- Assuming any increasing list is an AP → Check that the difference between every pair of consecutive terms is the same, not just the first pair.
- Confusing the position number n with the term value aₙ → The position tells you where the term sits in the sequence; the term value is the actual number at that position.
- Forgetting to subtract when finding the common difference → Always compute d as the later term minus the earlier term: d = a₂ − a₁, not a₁ − a₂.
- Substituting wrongly in the formula → Write out each step clearly: first compute (n − 1), then multiply by d, then add a.
Quick revision
- A sequence is an ordered list following a definite rule; each number is called a term.
- In an AP, every consecutive pair of terms has the same difference, called the common difference d.
- The nth term of an AP is aₙ = a + (n − 1) × d.
- To check for an AP, verify that all consecutive differences are equal.
- The general term formula lets you find any term directly without listing all previous terms.