CLASS 9 • MATHS • CHAPTER 8
Predicting What Comes Next: Sequences and Progressions
Short, colourful, handwritten-style notes by EduGrown
1What is a Sequence?
📝 Sequence & Term
A sequence is an ordered list of numbers. Each number in the list is called a term.
In the square numbers 1, 4, 9, 16, 25, … → 1 is the first term, 4 the second term, 25 the fifth term.
In the square numbers 1, 4, 9, 16, 25, … → 1 is the first term, 4 the second term, 25 the fifth term.
⭐ Finite or Infinite?
- Infinite sequence → goes on forever, shown by three dots …
e.g. 1, 2, 3, 4, 5, … - Finite sequence → has a fixed number of terms
e.g. 6, 12, 24, 48, 96 (only 5 terms)
💡 Why study sequences?
Patterns are everywhere — in nature, art, music and money. Sequences let us find the rule behind a pattern so we can predict what comes next.
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2Famous Number Patterns
| Sequence | Terms | Pattern |
|---|---|---|
| Natural numbers | 1, 2, 3, 4, 5, … | Add 1 each time |
| Odd numbers | 1, 3, 5, 7, 9, … | Add 2 each time |
| Triangular numbers | 1, 3, 6, 10, 15, … | Differences 2, 3, 4, 5… |
| Square numbers | 1, 4, 9, 16, 25, … | Differences 3, 5, 7, 9… |
⭐ Triangular numbers = sums of natural numbers
1 = 1 | 3 = 1 + 2 | 6 = 1 + 2 + 3 | 10 = 1 + 2 + 3 + 4 | 15 = 1 + 2 + 3 + 4 + 5
So each triangular number is the sum of natural numbers up to that position.
So each triangular number is the sum of natural numbers up to that position.
⭐ Square numbers = sums of odd numbers
1 = 1 | 4 = 1 + 3 | 9 = 1 + 3 + 5 | 16 = 1 + 3 + 5 + 7
Each square number is the sum of the odd numbers up to that term.
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Each square number is the sum of the odd numbers up to that term.
3Term Notation
📝 Naming the terms
We write t1 for the first term, t2 for the second term, and so on. The subscript matches the position.
For odd numbers: t1 = 1, t2 = 3, t3 = 5, t4 = 7 …
So t4 = 7 means ‘the term in the 4th position is 7’.
For odd numbers: t1 = 1, t2 = 3, t3 = 5, t4 = 7 …
So t4 = 7 means ‘the term in the 4th position is 7’.
💡 Many sequences at once
Use different letters for different sequences: t1, t2… for one, s1, s2… for another, u1, u2… for a third.
⚠️ Important point
In tn, the position n is always a natural number (1, 2, 3, …).
But the term itself can be a fraction, a negative number or any real number!
e.g. 1, 12, 13, 14, … or −7, −3, 1, 5, 9, …
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But the term itself can be a fraction, a negative number or any real number!
e.g. 1, 12, 13, 14, … or −7, −3, 1, 5, 9, …
4Explicit Rule for a Sequence
📝 Explicit formula
A rule that uses the position number n to directly calculate the value of the nth term.
✏️ Example
un = 2n − 1
u1 = 2(1) − 1 = 1, u2 = 2(2) − 1 = 3, u3 = 2(3) − 1 = 5
So this is the rule for the sequence of odd numbers.
u1 = 2(1) − 1 = 1, u2 = 2(2) − 1 = 3, u3 = 2(3) − 1 = 5
So this is the rule for the sequence of odd numbers.
⭐ Two big powers of the explicit rule
- Find any term instantly — the 53rd, the 300th, even the 1170th — without knowing the terms before it.
- Check if a number belongs to the sequence, and find its position.
✏️ Is 137 an odd-number term?
Solve 2n − 1 = 137 → 2n = 138 → n = 69
Yes! 137 is the 69th term.
Yes! 137 is the 69th term.
⚠️ When a number is NOT a term
For sn = 5n − 2, check 471: 5n − 2 = 471 → 5n = 473 → n = 94.6
Since n must be a natural number, 471 is not a term of this sequence.
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Since n must be a natural number, 471 is not a term of this sequence.
5Recursive Rule for a Sequence
📝 Recursive formula
A rule that gives a term by using the previous term(s). You must know the earlier terms to use it.
✏️ Example — the sequence 1, 4, 7, 10, 13, …
Each term is 3 more than the one before.
Explicit: tn = 3n − 2
Recursive: t1 = 1, tn = tn−1 + 3 (for n ≥ 2)
Explicit: tn = 3n − 2
Recursive: t1 = 1, tn = tn−1 + 3 (for n ≥ 2)
✏️ Example — u1 = 1, un = 2un−1 + 3
u2 = 2(1) + 3 = 5
u3 = 2(5) + 3 = 13
u4 = 2(13) + 3 = 29
First four terms: 1, 5, 13, 29
u3 = 2(5) + 3 = 13
u4 = 2(13) + 3 = 29
First four terms: 1, 5, 13, 29
💡 Explicit vs Recursive
Explicit = jump straight to any term. Recursive = build the sequence step by step.
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6Virahānka–Fibonacci Sequence
⭐ A rule that uses the previous TWO terms
V1 = 1, V2 = 2, Vn = Vn−1 + Vn−2 (for n ≥ 3)
1, 2, 3, 5, 8, 13, 21, 34, …
Each term = sum of the previous two.
📜 A proudly Indian discovery
First written down by Virahānka (7th century CE) in his work Vṛttajātisamuchaya — he found it while studying Prakrit poetry and metre!
Later studied by Gopāla (c. 1135 CE) and Hemachandra (c. 1150 CE), and much later by the Italian mathematician Fibonacci (c. 1200 CE).
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Later studied by Gopāla (c. 1135 CE) and Hemachandra (c. 1150 CE), and much later by the Italian mathematician Fibonacci (c. 1200 CE).
7Arithmetic Progression (AP)
📝 Arithmetic Progression
A sequence in which the difference between consecutive terms is constant.
That fixed number is called the common difference (d).
That fixed number is called the common difference (d).
⭐ General form of an AP
a, a + d, a + 2d, a + 3d, …
where a = first term and d = common difference.
✏️ From the picture
At each stage 4 squares get added at the corners:
1, 1 + 4, 1 + 2×4, 1 + 3×4, … 1, 5, 9, 13, 17, 21, … Here a = 1 and d = 4, so tn = 1 + (n − 1)×4 = 4n − 3
1, 1 + 4, 1 + 2×4, 1 + 3×4, … 1, 5, 9, 13, 17, 21, … Here a = 1 and d = 4, so tn = 1 + (n − 1)×4 = 4n − 3
💡 d can be negative too!
In 11, 7, 3, −1, −5, … the numbers go down by 4.
So a = 11 and d = −4 — still a perfectly good AP.
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So a = 11 and d = −4 — still a perfectly good AP.
8nth Term of an AP & Its Graph
⭐ The two rules for every AP
Explicit: tn = a + (n − 1) d
Recursive: t1 = a, tn = tn−1 + d (n ≥ 2)
| Stage number | 1 | 2 | 3 | 4 | 5 | … | n |
|---|---|---|---|---|---|---|---|
| Number of squares | 1 | 5 | 9 | 13 | 17 | … | 4n − 3 |
💡 The visual signature of an AP
Plot the points (stage, value): (1, 1), (2, 5), (3, 9), (4, 13), (5, 17).
They always lie on a perfect straight line — that is how you spot an AP on a graph!
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They always lie on a perfect straight line — that is how you spot an AP on a graph!
9Sum of the First n Natural Numbers
✏️ The clever trick
Write the sum forwards, then backwards, and add:
S = 1 + 2 + 3 + … + 10
S = 10 + 9 + 8 + … + 1
Every column adds to 11, and there are 10 columns.
So 2S = 11 × 10 = 110 → S = 55
S = 1 + 2 + 3 + … + 10
S = 10 + 9 + 8 + … + 1
Every column adds to 11, and there are 10 columns.
So 2S = 11 × 10 = 110 → S = 55
⭐ The formula
Sn = 1 + 2 + 3 + … + n = n(n + 1)2
The picture proof: two copies of the staircase fit together into an n × (n+1) rectangle.
💡 Same formula, second job
The nth triangular number is exactly this sum:
tn = n(n + 1)2
✏️ Sum of a middle stretch
25 + 26 + … + 58 = S58 − S24
= 58×592 − 24×252 = 1711 − 300 = 1411
= 58×592 − 24×252 = 1711 − 300 = 1411
📜 Did you know?
The first known written mention of this result is in Āryabhaṭa’s Āryabhaṭīya, Chapter 2, Verse 19.
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10Geometric Progression (GP)
📝 Geometric Progression
A sequence in which each term is obtained by multiplying the previous term by a fixed number.
That fixed multiplier is the common ratio (r).
That fixed multiplier is the common ratio (r).
⭐ General form of a GP
a, ar, ar2, ar3, …, arn−1
where a = first term and r = common ratio.
✏️ Checking the ratio
For 3, 6, 12, 24, 48, 96:
63 = 126 = 2412 = 4824 = 9648 = 2
All ratios equal → it is a GP with a = 3, r = 2.
63 = 126 = 2412 = 4824 = 9648 = 2
All ratios equal → it is a GP with a = 3, r = 2.
💡 How to test any sequence
Divide each term by the one before it. If you always get the same number, it is a GP — and that number is r.
r can be a fraction (like 34) or negative (like −1)!
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r can be a fraction (like 34) or negative (like −1)!
11nth Term of a GP & Its Graph
⭐ The two rules for every GP
Explicit: tn = a rn−1
Recursive: t1 = a, tn = r × tn−1 (n ≥ 2)
| Stage number | 1 | 2 | 3 | 4 | 5 | … | n |
|---|---|---|---|---|---|---|---|
| Number of squares | 3 | 6 | 12 | 24 | 48 | … | 3 × 2n−1 |
💡 The visual signature of a GP
Plotting (1, 3), (2, 6), (3, 12), (4, 24), (5, 48) gives a curve that shoots upwards — never a straight line.
🔸 If r > 1, the curve rises very fast. If 0 < r < 1, it falls towards 0.
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🔸 If r > 1, the curve rises very fast. If 0 < r < 1, it falls towards 0.
12Fun with Fractals — Sierpiński Triangle
📝 How it is made
Start with a black equilateral triangle. Join the midpoints of the three sides → four small triangles. Remove the middle one. Now repeat on every black triangle… forever!
| Stage (n) | 0 | 1 | 2 | 3 | 4 | 5 | n |
|---|---|---|---|---|---|---|---|
| Black triangles | 1 = 30 | 3 = 31 | 9 = 32 | 27 = 33 | 81 | 243 | 3n |
| Black area | 1 | 34 | (34)2 | (34)3 | (34)4 | (34)5 | (34)n |
⭐ Two GPs hiding in one fractal
- Number of triangles: tn = 3n (r = 3, grows fast)
- Black area: sn = (34)n (r = 34, shrinks towards 0)
13Fractals in Nature
📝 What is a fractal?
A shape or pattern that repeats itself at different scales — zoom in on a small part and it looks like the whole thing!
💡 Where else?
Broccoli, coastlines, river networks, lightning and even blood vessels! Fractals are built from simple repeated rules yet look wonderfully complex — and they always lead to geometric progressions.
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14Real-life Examples of AP and GP
✏️ AP — the taxi fare
Booking fee ₹200 + ₹40 per km.
After 1 km: ₹240 | 2 km: ₹280 | 3 km: ₹320 240, 280, 320, … An AP with a = 240, d = 40.
Fare for n km = 240 + (n − 1)×40 = 200 + 40n
After 1 km: ₹240 | 2 km: ₹280 | 3 km: ₹320 240, 280, 320, … An AP with a = 240, d = 40.
Fare for n km = 240 + (n − 1)×40 = 200 + 40n
✏️ GP — the bouncing ball
A ball is dropped from 24 feet and each bounce reaches 34 (0.75) of the previous height.
18 → 13.5 → 10.125 → 7.594 → 5.695 …
A GP with a = 18 and r = 0.75. Since r < 1, the heights keep shrinking.
18 → 13.5 → 10.125 → 7.594 → 5.695 …
A GP with a = 18 and r = 0.75. Since r < 1, the heights keep shrinking.
💡 Spot them around you
AP: yearly salary increments, seats increasing row by row, simple interest.
GP: bacteria doubling every hour, compound interest, halving of medicine in the body.
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GP: bacteria doubling every hour, compound interest, halving of medicine in the body.
15AP vs GP — All Formulas
| Arithmetic Progression (AP) | Geometric Progression (GP) | |
|---|---|---|
| How it grows | Add a fixed number d | Multiply by a fixed number r |
| Key number | Common difference d | Common ratio r |
| How to find it | t2 − t1 | t2 ÷ t1 |
| General form | a, a+d, a+2d, a+3d, … | a, ar, ar2, ar3, … |
| Explicit rule | tn = a + (n − 1)d | tn = a rn−1 |
| Recursive rule | tn = tn−1 + d | tn = r × tn−1 |
| Graph shape | Straight line | Curve (never straight) |
| Example | 1, 5, 9, 13, … | 3, 6, 12, 24, … |
⭐ Two more must-know formulas
1 + 2 + 3 + … + n = n(n + 1)2
nth triangular number = n(n + 1)2
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16Common Mistakes to Avoid
❌ Don’t do this
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- Writing tn = a + nd for an AP. The correct rule is a + (n − 1)d.
- Writing tn = a rn for a GP. The correct rule is a rn−1.
- Calling a sequence an AP after checking only one pair of terms — check several differences.
- Forgetting that d and r can be negative or fractions.
- Accepting a decimal value of n as an answer. Position n must be a natural number, so a decimal means the number is not a term.
- Mixing up explicit and recursive rules — recursion always needs the previous term.
- Assuming every sequence has a neat rule. Prime numbers 2, 3, 5, 7, 11, … have no simple pattern!
- Using n(n+1)2 for the sum of any AP — it is only for 1 + 2 + 3 + … + n.
17Chapter Summary — Quick Revision
🏆 All of Chapter 8 in 10 lines
- A sequence is an ordered list of numbers; each number is a term. It may be finite or infinite.
- An explicit formula uses the position n to find the term directly; a recursive formula uses previous terms.
- Triangular numbers 1, 3, 6, 10, 15, … have nth term n(n+1)2, which is also the sum 1 + 2 + … + n.
- Square numbers are sums of odd numbers; triangular numbers are sums of natural numbers.
- The Virahānka–Fibonacci sequence 1, 2, 3, 5, 8, 13, … adds the previous two terms.
- An AP adds a fixed common difference d: tn = a + (n − 1)d. Its points lie on a straight line.
- A GP multiplies by a fixed common ratio r: tn = a rn−1. Its points lie on a curve.
- Find d by subtracting consecutive terms; find r by dividing consecutive terms.
- Fractals like the Sierpiński triangle and square carpet contain hidden geometric progressions.
- APs and GPs are everywhere — taxi fares, salary hikes, bouncing balls and bacteria growth.
✨ Happy Learning — @edugrown ✨