Schooling · Class 9 · Mathematics · Predicting What Comes Next: Exploring Sequences and Progressions · Chapter on a page
CBSE Class 9 · Mathematics · NCERT Ganita Manjari
Predicting What Comes Next: Exploring Sequences and Progressions: chapter on a page
A cut of Shishya's notes for this chapter: the first paragraph of the introduction and the key ideas, formulas and facts, common mistakes and quick revision lists, word for word. Nothing is added or reworded; the worked examples are left out (they are on the chapter page).
About the notes: Written by Shishya's AI on 26 Sept 2026 from the chapter's title and class level, in Shishya's own words — not a copy or summary of the textbook. Read the official chapter for the book's own text, activities and exercises.
This note's current text has no applied fact-check on record.
Mind map
Drawn from the same notes by a fixed rule (no AI): each label is the item's own words — its bold term, its label before a colon, a short formula as written, or its first words.
The mind map as a list
- Key ideas
- sequence
- term
- rule
- arithmetic progression (AP)
- In an AP with first term a and…
- Not every list of numbers is a…
- Recognising whether differences…
- Formulas and facts
- nth term of an AP
- Common difference
- General term of a sequence
- Checking for AP
- First term
- Watch out for
- Writing the nth term formula as aₙ =…
- Assuming any increasing list is an AP
- Confusing the position number n with…
- Forgetting to subtract when finding…
- Substituting wrongly in the formula
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.
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.
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.