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MathematicsClass 9Predicting What Comes Next: Exploring Sequences and Progressions

Predicting What Comes Next: Exploring Sequences and Progressions | Class 9 Mathematics Notes

By ConceptScroll Team · Published on 17 July 2026 · 4 min read

Predicting What Comes Next: Exploring Sequences and Progressions – this guide gives you a concise, exam-ready overview of Predicting What Comes Next: Exploring Sequences and Progressions from Class 9 Mathematics, written by ConceptScroll editors and reviewed against the latest NCERT textbook.

8.1 INTRODUCTION TO SEQUENCES

Sequences are ordered lists of numbers or objects arranged in a particular order, forming patterns that help us understand and predict what comes next. In everyday life, patterns are everywhere—in nature, art, music, and finance. Mathematics formalizes these patterns as sequences, which can be finite or infinite. For example, natural numbers (1, 2, 3, 4, 5, 6, ...) and odd numbers (1, 3, 5, 7, 9, 11, ...) are infinite sequences. The sequence of triangular numbers (1, 3, 6, 10, 15, 21, ...) and square numbers (1, 4, 9, 16, 25, 36, ...) are also infinite, each term representing a special pattern or sum. The notation t₁, t₂, t₃, ... is used to denote the first, second, third, and so on terms of a sequence, connecting the position of a term to its value. Sequences can have terms that are positive, negative, fractions, or any real numbers. Understanding sequences allows us to explore how numbers grow, shrink, or repeat, and apply these ideas to solve real-life problems.

📊 Diagram: Fig. 8.1 shows the first five triangular numbers represented as triangular arrays of dots, illustrating how each number forms a triangle shape with dots.

🧪 Activity: Think and Reflect: Describe the pattern in each given sequence and predict the next few numbers.

🔗 Connection: This section introduces sequences and notation, setting the foundation for understanding explicit and recursive rules in the next sections.

Frequently asked questions

Consider the sequence 1, 4, 7, 10, 13, ... Can you predict the next four terms? Can you derive the first 10 terms of the sequence obtained by adding all the terms up to a given term of this sequence? (Hint: The first term is 1. The second term is $1 + 4 = 5$, the third term is $1 + 4 + 7 = 12$, and so on.)

Step 1: Identify the pattern in the sequence 1, 4, 7, 10, 13, ... The sequence increases by 3 each time, so it is an arithmetic progression (AP) with first term a = 1 and common difference d = 3.

Step 2: Predict the next four terms: 6th term = 13 + 3 = 16 7th term = 16 + 3 = 19 8th term = 19 + 3 = 22 9th term = 22 + 3 = 25

So, the next four terms are 16, 19, 22, 25.

Step 3: Derive the first 10 terms of the sequence obtained by adding all terms up to a given term. This is the sequence of parti

Can you write $t_5, t_6, t_7$ and $t_8$ for the sequence of triangular numbers?

The sequence of triangular numbers is given by the formula $t_n = \frac{n(n+1)}{2}$. Using this, we calculate:

$t_5 = \frac{5 \times 6}{2} = \frac{30}{2} = 15$

$t_6 = \frac{6 \times 7}{2} = \frac{42}{2} = 21$

$t_7 = \frac{7 \times 8}{2} = \frac{56}{2} = 28$

$t_8 = \frac{8 \times 9}{2} = \frac{72}{2} = 36$

Thus, $t_5=15$, $t_6=21$, $t_7=28$, and $t_8=36$.

Using the explicit rule $u_n = 2n - 1$, find the $53^{\text{rd}}$ term, the $108^{\text{th}}$ term, and the $1170^{\text{th}}$ term of the odd number sequence.

Given the explicit rule $u_n = 2n - 1$, we find the terms as follows:

(i) For the 53rd term: $u_{53} = 2 \times 53 - 1 = 106 - 1 = 105$

(ii) For the 108th term: $u_{108} = 2 \times 108 - 1 = 216 - 1 = 215$

(iii) For the 1170th term: $u_{1170} = 2 \times 1170 - 1 = 2340 - 1 = 2339$

Thus, the 53rd term is 105, the 108th term is 215, and the 1170th term is 2339.

Can you find the rule describing the $n^{th}$ term of the sequence of square numbers?

The sequence of square numbers is: 1, 4, 9, 16, 25, ... The nth term of this sequence is given by $u_n = n^2$. This means the term at position n is the square of n.

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