Chapter 3 now focuses on spotting patterns, writing explicit and recursive rules,
understanding Arithmetic Progressions and Geometric Progressions,
and connecting them to fractals and the Tower of Hanoi puzzle.
Sequences
Recursive Rules
AP and nth Term
GP, Fractals, Hanoi
Explicit Rule\( a_n = 4n + 1 \)
Recursive Rule\( a_n = a_{n-1} + 3 \)
ProgressionsAP adds. GP multiplies.
The official CBSE Class IX 2026-27 syllabus places this chapter in Algebra and lists: introduction
to sequences, explicit or general rule, recursive rule, Arithmetic Progressions, sum of the first
\( n \) natural numbers, Geometric Progressions, applications of GP in fractals, and the Tower of
Hanoi puzzle.
Syllabus Snapshot
What This Chapter Wants You to Do
This chapter is about pattern thinking. You should be able to observe a pattern, write
a rule for it, generate terms, and decide whether it grows by adding or by multiplying.
Introduction to sequences
Explicit rule
Recursive rule
Arithmetic Progressions
nth term of AP
Sum of first \( n \) natural numbers
Geometric Progressions
Fractals and GP
Tower of Hanoi
Observe
Recognise the pattern in a list of numbers and predict the next few terms.
Represent
Write both recursive and explicit rules for sequences and use them to generate terms.
Generalise
Identify AP and GP, find the nth term, and interpret growth patterns in real contexts.
Compute
Use computational thinking to build step-by-step rules and solve the Tower of Hanoi puzzle.
Core Idea
1. What Is a Sequence?
A sequence is an ordered list of numbers arranged according to some rule.
Order matters. Each number in the list is called a term.
\( 3, 6, 9, 12, 15, \ldots \)This is a sequence because each term is obtained by following a pattern.
If the same change happens again and again, look for an
AP.
If the same multiplication happens again and again, look
for a GP.
To find the next term, first decide whether the pattern is
additive or multiplicative.
Explicit or General Rule
A formula that gives the nth term directly. Example: \( a_n = 2n + 1 \).
Recursive Rule
A rule that tells how one term is built from the previous term. Example:
\( a_n = a_{n-1} + 2 \).
Rule Lab
2. Explicit Rule and Recursive Rule of a Sequence
The explicit rule jumps directly to the required term. The recursive rule moves step by step from one
term to the next.
Sequence Builder
Change the starting term and the fixed change to build a number sequence.
Rules
Explicit: \( a_n = 2 + (n-1)\cdot 3 \)
Recursive: \( a_1 = 2,\; a_n = a_{n-1} + 3 \)
Pattern Story
Start from 2 and add 3 again and again.
First Six Terms
n
\( a_n \)
Tip: explicit rule uses \( n \), recursive rule uses the previous term.
Arithmetic Progression
3. Arithmetic Progressions (AP)
An Arithmetic Progression is a sequence where the difference between consecutive terms
is constant. This constant is called the common difference \( d \).
\( a,\; a+d,\; a+2d,\; a+3d,\ldots \)nth term of an AP: \( a_n = a + (n-1)d \)
AP Explorer
\( a_n = 5 + (n-1)\cdot 4 \)
For \( n = 6 \), the term is 25.
AP test: subtract consecutive terms.
If the difference is constant: it is an AP.
Real contexts: staircase blocks, weekly savings, page numbers, fixed marks increase.
Visualising an AP
First Term5
Difference4
nth Term25
Special AP: Sum of the First \( n \) Natural Numbers
The natural numbers \( 1, 2, 3, 4, \ldots \) form an AP with first term 1 and common difference 1.
Their sum is
\( 1 + 2 + \cdots + n = \frac{n(n+1)}{2} \).
Use the formula when counting totals in rows, seats, or stacked objects.
Geometric Progression
4. Geometric Progressions (GP)
A Geometric Progression is a sequence where every term is obtained by multiplying the
previous term by the same number. That fixed multiplier is called the common ratio
\( r \).
\( a,\; ar,\; ar^2,\; ar^3,\ldots \)nth term of a GP: \( a_n = ar^{n-1} \)
GP Explorer
\( a_n = 2\cdot 2^{n-1} \)
For \( n = 5 \), the term is 32.
Practical Contexts
Doubling bacteria, branching trees, repeated folding, and powers of 2 often create GPs.
Visualising a GP
First Term2
Ratio2
nth Term32
GP in Fractals
5. Applications of GP in Fractals
Fractals are shapes that repeat a pattern at smaller and smaller scales. This repeated scaling often
follows a GP.
Fractal Stage Explorer
Think of a simple branch pattern where each stage creates twice as many pieces, each one-third as long.
Number of pieces: \( 2^3 = 8 \)
Length of each piece: \( \left(\frac{1}{3}\right)^3 = \frac{1}{27} \) of the original.
Piece count often forms a GP like \( 1,2,4,8,\ldots \)
Piece length often forms a GP like \( 1,\frac13,\frac19,\frac1{27},\ldots \)
Fractal Visual
Each stage repeats the same growth rule at a smaller scale.
Puzzle Thinking
6. Tower of Hanoi Puzzle
In the Tower of Hanoi puzzle, all disks begin on one peg and must be moved to another peg using a third
peg, while never placing a larger disk on a smaller one.
\( T_n = 2T_{n-1} + 1 \)Minimum number of moves for \( n \) disks: \( 2^n - 1 \)
Hanoi Move Counter
Minimum moves: \( 2^4 - 1 = 15 \)
Recursive relation: move 3 disks, move the largest disk, then move 3 disks again.
Rule 1: Move only one disk at a time.
Rule 2: A larger disk can never sit on a smaller disk.
Rule 3: Break the big problem into smaller copies of the same problem.
Starting Position
Source
Helper
Target
Computational Thinking
7. How to Model a Pattern Like a Computer
Computational thinking means turning a pattern into a clear step-by-step rule that can always be
repeated.
Look for change. Compare consecutive terms.
Decide the type. Equal difference suggests AP. Equal ratio suggests GP.
Write a rule. Choose explicit if you want the nth term directly. Choose recursive if you want the next term from the previous one.
Test the rule. Generate 3 or 4 terms to see whether the rule works.
Connect to context. Ask what the numbers represent in a real situation.
Example
If a game gives 10 points in level 1 and 15 points in level 2, then 20, 25, 30, the pattern adds 5
each time. So it is an AP with explicit rule \( a_n = 10 + (n-1)5 \) and recursive rule
\( a_n = a_{n-1} + 5 \).
Quick Check
8. Exam Smart Zone
Question 1
Which rule is recursive?
A recursive rule mentions the previous term.
Question 2
What is the 5th term of the AP \( 3, 7, 11, 15, \ldots \)?
Add the common difference 4 again and again.
Question 3
Which sequence is a GP?
In a GP, each term is multiplied by the same number.
Final Revision
9. Final Self-check
I can identify the pattern in a sequence and predict the next terms.
I can write both explicit and recursive rules.
I can recognise AP and find its nth term.
I can recognise GP and find its nth term.
I can connect GPs to fractals and repeated scaling.
I can explain why Tower of Hanoi uses recursion and \( 2^n - 1 \).