New NCERT Ganita Manjari · Class 9 Maths · Part I · Chapter 8: Predicting What Comes Next: Exploring Sequences and Progressions
This chapter is about sequences, ordered lists of numbers that follow a pattern. It explains explicit and recursive rules, the Virahānka–Fibonacci sequence, arithmetic progressions (AP) and geometric progressions (GP). It also shows how to add the first n natural numbers and how AP and GP graphs look different.
From the Class 9 Maths Formula Sheet, made as per the new NCERT book Ganita Manjari · Pages 42–47 of the book.
Key points
- A sequence is an ordered list of numbers; each number is a term. It can be finite or infinite.
- tₙ means the nth term; the subscript n is the position and is always a natural number.
- An explicit rule finds a term directly from its position n; a recursive rule uses the previous term(s) and a starting value.
- A number is a term of a sequence only if the n you get is a natural number.
- In the Virahānka–Fibonacci sequence each term is the sum of the previous two: 1, 2, 3, 5, 8, 13, 21, …
- In an AP the difference between consecutive terms is constant; this is the common difference d.
- The points of an AP lie on a straight line when plotted.
- The sum of the first n natural numbers is n(n + 1)/2, which is also the nth triangular number.
- In a GP each term is the previous term multiplied by a fixed number r, the common ratio.
- The points of a GP lie on a curve that rises faster and faster when r > 1, not on a line.
- The Sierpiński triangle is a fractal: the number of black triangles is a GP with r = 3 and the black area is a GP with r = 3/4.
- 1 + 3 + 5 + … (n odd numbers) = n².
Formulas
| nth term of an AP | tₙ = a + (n − 1)d a = first term, d = t₂ − t₁ |
| Recursive rule of an AP | t₁ = a, tₙ = tₙ₋₁ + d for n ≥ 2 |
| nth term of a GP | tₙ = arⁿ⁻¹ a = first term, r = t₂/t₁ |
| Recursive rule of a GP | t₁ = a, tₙ = r × tₙ₋₁ for n ≥ 2 |
| Sum of first n natural numbers | Sₙ = 1 + 2 + 3 + … + n = n(n + 1)/2 also the nth triangular number |
| Āryabhaṭa's sum rule | sum = (first term + last term)/2 × number of terms |
| Sum of first n odd numbers | 1 + 3 + 5 + … (n odd numbers) = n² |
| Virahānka–Fibonacci sequence | V₁ = 1, V₂ = 2, Vₙ = Vₙ₋₁ + Vₙ₋₂ for n ≥ 3 |
| Odd numbers and squares | uₙ = 2n − 1; tₙ = n² |
| Sierpiński triangle | tₙ = 3ⁿ triangles, black area sₙ = (3/4)ⁿ stages counted from 0 |
Diagrams to learn
- Table of familiar sequences (natural, odd, triangular, square numbers)
- Triangular numbers dot pattern and odd-number square
- Short and long syllable rhythms for the Virahānka sequence
- AP square pattern stages and straight-line graph
- Dot diagram for the sum of first n natural numbers
- GP square stages and curved graph
- Sierpiński triangle stages with graphs
- Bouncing ball heights bar graph
- AP or GP comparison table
- A number is a term only if n comes out as a natural number; there is no '94.6th position'.
- tₙ = a + nd is wrong. The first term has no d added, so the nth term has only (n − 1) differences: tₙ = a + (n − 1)d.
- d = (later term) − (earlier term), so it can be negative.
- tₙ = arⁿ is wrong; the power is n − 1. 1, 4, 9, 16, … is neither an AP nor a GP, while 5, 5, 5, … is both.
A real page from the book
Get the full chapter in the book
This page is a summary. The Class 9 Maths Formula Sheet has the complete chapter in colour, with every diagram and worked example.
Frequently asked questions
What is the formula for the nth term of an AP?
tₙ = a + (n − 1)d, where a is the first term and d is the common difference. For 7, 13, 19, …, the term 121 is the 20th term.
What is the difference between an AP and a GP?
In an AP you add a fixed number d to get the next term, and the points lie on a straight line. In a GP you multiply by a fixed number r, and the points lie on a curve.
What is the sum of the first n natural numbers?
1 + 2 + 3 + … + n = n(n + 1)/2. For example, the sum of the first 100 natural numbers is 100 × 101/2 = 5050.
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