NCERT Solutions for Class 9th Maths Chapter 8 Predicting What Comes Next: Exploring Sequences
Updated on 2026-09-19
About this chapter
A sequence is an ordered list of numbers; each number is a term . The notation t 1 , t 2 , t 3 , … ties a term to its position, so t 4 = 7 says "the term in the 4th place is 7". Sequences may be finite (6, 12, 24, 48, 96) or infinite (1, 2, 3, 4, …). An explicit rule computes t n straight from the position n — t n = 2n – 1 gives the 1170th odd number without listing the first 1169. It also runs backwards: solving t n = k tells you whether k is a term, and where. The answer counts only if n comes out a natural number. A recursive rule gives a term from earlier terms, e.g. t 1 = 1, t n = t n–1 + 3. It may reach back further than one step: the Virahānka–Fibonacci sequence uses V n = V n–1 + V n–2 , giving 1, 2, 3, 5, 8, 13, 21, 34, … It was set down by Virahānka in the 7th century CE in the V
- Introduction to Sequences
- Explicit Rule for a Sequence
- Recursive Rule for a Sequence
- Arithmetic Progressions
- Visualising an AP
- Sum of the First n Natural Numbers
- Geometric Progressions
- Fun with Fractals
- Chapter 8 review (questions marked * are the harder set)
Quick revision
| Idea | In symbols | What it says | Where students slip |
|---|---|---|---|
| Term and position | tn | n is the place, tn is the value sitting there | Reading t5 = 9 as 'the term 5 equals 9' |
| Explicit rule | tn = f(n) | Jump to any term directly from n | Using it without checking that n is a natural number |
| Recursive rule | t1 given, tn from tn–1 | Each term is built from the ones before it | Forgetting to state the starting term |
| Is k a term? | solve tn = k | k is a term only if n is a natural number | Accepting n = 94.6 as an answer |
| AP | tn = a + (n – 1)d | Add the same d each step | Using n instead of (n – 1) |
| Common difference | d = tn – tn–1 | Later term minus earlier term; may be negative | Subtracting the wrong way round in a falling AP |
| GP | tn = arn–1 | Multiply by the same r each step | Writing arn instead of arn–1 |
| Common ratio | r = tn ÷ tn–1 | Every consecutive pair must give the same r | Checking only one pair of terms |
| Sum 1 + 2 + … + n | Sn = n(n + 1)/2 | Pair the ends: n pairs each worth (n + 1), halved | Dropping the division by 2 |
| Triangular numbers | 1, 3, 6, 10, 15, … | tn = n(n + 1)/2, the nth partial sum of 1, 2, 3, … | Confusing them with square numbers |
| Square numbers | 1, 4, 9, 16, … | tn = n2 = 1 + 3 + 5 + … + (2n – 1) | Missing that the gaps are the odd numbers |
| AP graph vs GP graph | line vs curve | Constant addition is linear; constant multiplication is not | Calling any rising sequence an AP |
Exercises
- Think and Reflect — Introduction to Sequences Page 174
- In-text Questions — Introduction to Sequences Page 174–175
- In-text Questions — Introduction to Sequences Page 176
- Think and Reflect — Introduction to Sequences Page 176
- Think and Reflect — Explicit Rule for a Sequence Page 177
- In-text Questions — Explicit Rule for a Sequence Page 177–178
- In-text Questions — Recursive Rule for a Sequence Page 179
- Exercise Set 8.1 — Recursive Rule for a Sequence Page 179–180
- Think and Reflect — Arithmetic Progressions Page 180
- Think and Reflect — Arithmetic Progressions Page 181
- In-text Questions — Visualising an AP Page 182–183
- Think and Reflect — Sum of the First n Natural Numbers Page 184
- Think and Reflect — Sum of the First n Natural Numbers Page 185
- Exercise Set 8.2 — Sum of the First n Natural Numbers Page 185–186
- Think and Reflect — Geometric Progressions Page 186
- In-text Questions — Geometric Progressions Page 187
- Think and Reflect — Fun with Fractals Page 189
- In-text Questions — Fun with Fractals Page 189
- Exercise Set 8.3 — Geometric Progressions (questions marked * are the harder set) Page 193–194
- Chapter 8 review (questions marked * are the harder set) — End-of-Chapter Exercises Page 194–195