Chapter 8 — Predicting What Comes Next: Exploring Sequences and Progressions
Open PDFReads in your browser→Summary
Chapter 8 of NCERT Class 9 Maths, "Predicting What Comes Next: Exploring Sequences and Progressions", introduces sequences as ordered lists of numbers and teaches explicit and recursive rules, arithmetic progressions, geometric progressions, and the sum of the first n natural numbers.
This chapter explores patterns in number sequences and the rules used to predict their terms. It defines a sequence as an ordered list of numbers, each called a term, and shows how to write explicit rules (using the position n, like un = 2n - 1) and recursive rules (relating each term to previous ones, like the Virahanka-Fibonacci sequence). It then studies arithmetic progressions (constant common difference d, nth term a + (n - 1)d) and geometric progressions (constant common ratio r, nth term a*r^(n-1)). It also derives the sum of the first n natural numbers, n(n + 1)/2, and links GPs to fractals such as the Sierpinski triangle.
Key points & formulas
- 01A sequence is an ordered list of numbers; each number is a term
- 02An explicit rule uses position n to compute the nth term directly
- 03A recursive rule defines each term from previous terms
- 04Virahanka-Fibonacci: V1 = 1, V2 = 2, Vn = V(n-1) + V(n-2)
- 05AP nth term: tn = a + (n - 1)d, with common difference d
- 06GP nth term: tn = a*r^(n-1), with common ratio r
- 07Sum of first n natural numbers: Sn = n(n + 1)/2
Frequently asked questions
01What is the difference between an explicit rule and a recursive rule for a sequence?
An explicit rule uses the term's position number n to calculate its value directly, such as un = 2n - 1, so you can find any term without knowing previous terms. A recursive rule defines a term using earlier terms, such as t1 = 1 and tn = t(n-1) + 3, so you must know previous terms to find the next ones.
02What is an arithmetic progression and what is its nth term formula?
An arithmetic progression (AP) is a sequence in which the difference between consecutive terms is constant; this fixed difference d is the common difference. The nth term of an AP is tn = a + (n - 1)d, where a is the first term, giving the general form a, a + d, a + 2d, a + 3d, and so on.
03What is a geometric progression and how is it different from an AP?
A geometric progression (GP) is a sequence in which each term after the first is obtained by multiplying the previous term by a fixed number called the common ratio r. Its nth term is tn = a*r^(n-1). Unlike an AP, where terms change by adding a constant difference, a GP changes by multiplying by a constant ratio.
04What is the formula for the sum of the first n natural numbers?
The sum of the first n natural numbers is Sn = n(n + 1)/2, derived by writing the sum forwards and backwards and adding the two. For example, the sum of the first 10 natural numbers is 55. This is also the nth triangular number.
More chapters in Ganita Manjari
This is the complete Ganita Manjari Chapter 8 as published by NCERT — every diagram, solved example, and exercise included, free. Browse all NCERT Class 9 textbooks.
Read offline with notes, solutions & mock tests
CBSE Prepmaster — free on iOS & Android