Complete visual guide to sequences, Arithmetic Progressions (AP), Geometric Progressions (GP), nth term, sum formulas, graphs, fractals, and the Tower of Hanoi. Entirely new to Class 9 in 2026-27. Chapter 6 Β· 7 Marks.
| Type | How to find next term | Example |
|---|---|---|
| Recursive | Each term defined using previous term(s) | aβ = aβββ + 3 |
| Explicit | nth term defined directly using n | aβ = 2n + 1 |
| General Term | Formula for any term β most useful! | Tβ = a + (nβ1)d |
Each term is obtained by adding a fixed number to the previous term.
Common difference d = 3 (constant addition)
Each term is obtained by multiplying a fixed number to the previous term.
Common ratio r = 3 (constant multiplication)
a=2, r=2. Doubles each time. Exponential growth.
a=64, r=Β½. Halves each time β approaches 0.
A fractal is a pattern that repeats itself at every scale. The SierpiΕski Triangle is a classic fractal built using an AP/GP pattern.
At each step, each triangle splits into 3 smaller ones. The number of triangles forms a GP: 1, 3, 9, 27, β¦ with first term 1 and r = 3. After n steps: 3βΏ triangles.
The Tower of Hanoi puzzle with n discs requires a minimum of 2βΏ β 1 moves. This forms a sequence of minimum moves:
The moves sequence 1, 3, 7, 15, 31 is actually 2βΏ β 1. It's related to a GP: the differences form a GP (1, 2, 4, 8, β¦). The legend says a monk moving 64 golden discs would take 2βΆβ΄β1 β 18 quintillion moves!
For the AP: 2, 7, 12, 17, β¦ find the 31st term.
Find the sum: 1 + 3 + 5 + 7 + β¦ + 99 (sum of first 50 odd numbers)
For the GP 5, 25, 125, β¦ find the 6th term.
The 4th term of a GP is 8 and the 8th term is 128. Find the first term and common ratio.
Common difference d is always the LATER term minus the EARLIER term. d = TββTβ. If you reverse it, you'll get the wrong sign.
The nth term equals the sum of n terms minus the sum of (nβ1) terms. This is tested in 2-mark questions β don't forget it.
Common ratio r = any term Γ· previous term. Always check r by trying Tβ/Tβ = Tβ/Tβ.
If three numbers are in AP, let them be (aβd), a, (a+d). Their sum = 3a and product is easy. Saves algebra.
If three numbers are in GP, let them be a/r, a, ar. Their product = aΒ³. Saves algebra enormously (as in Example 8).
This is Sβ for AP: a=1, d=1. Formula: n(n+1)/2. Use directly when asked to sum 1+2+3+β¦+n.
1+3+5+β¦+(2nβ1) = nΒ². Pure AP: a=1, d=2. Verified: 1+3+5=9=3Β². Very frequent in CBSE.
Plot AP terms β straight line (constant slope = d). Plot GP terms β exponential curve. CBSE asks to identify the type from graph.
For n discs: 1 disc β 1 move, 2 discs β 3 moves, 3 discs β 7 moves. General: 2βΏ β 1. Comes up in case-based questions.
When you know first and last term, use Sβ = n/2(a+l). When you know a and d but not l, use Sβ = n/2[2a+(nβ1)d].
| Concept | Formula | Notes |
|---|---|---|
| AP β nth term | Tβ = a + (nβ1)d | a=first term, d=common diff |
| AP β last term | l = a + (nβ1)d | Same formula with l instead of Tβ |
| AP β Sum (using d) | Sβ = n/2[2a+(nβ1)d] | Use when d is known |
| AP β Sum (using l) | Sβ = n/2(a+l) | Use when last term l is known |
| nth term from sum | Tβ = Sβ β Sβββ | For n β₯ 2; Tβ = Sβ |
| GP β nth term | Tβ = aΒ·rβΏβ»ΒΉ | a=first term, r=common ratio |
| Sum 1 to n | n(n+1)/2 | AP: a=1, d=1 |
| Sum first n odds | nΒ² | AP: a=1, d=2 |
| SierpiΕski triangles | 3βΏ at step n | GP with r=3 |
| Tower of Hanoi | 2βΏ β 1 for n discs | Related to GP |