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Lesson 1 of 8

Predicting What Comes Next: Exploring Sequences and Progressions · Lesson 1 of 8

Introduction to Sequences

“A sequence is a number pattern that remembers who comes next.”

Learning Objectives

• Understand a sequence as an ordered list. • Identify terms and term positions. • Distinguish finite and infinite sequences. • Recognise familiar numerical patterns. • Use notation such as t₁, t₂ and tₙ correctly.

Sequences

Patterns are all around us. We notice them in repeated music beats, tiles arranged on a floor, rows of seats in a hall, dates on a calendar, regular increases in savings, and even in many shapes found in nature. Some patterns repeat, while others grow or decrease in a predictable way. Mathematics helps us describe and study these patterns in an organised form.

One important way to represent a numerical pattern is through a sequence. A sequence is an ordered list of numbers written according to a particular rule or pattern. The word ordered is important because the position of each number matters. Changing the order can change the sequence completely. By studying how one term follows another, we can identify patterns, predict later terms, and understand the rule that connects the numbers.

Definition
Sequence

An ordered list of numbers in which each number is called a term.

Familiar Sequences

Problem
Look at these sequences: 1,2,3,4,… ; 1,3,5,7,… ; 1,3,6,10,… ; 1,4,9,16,… . What is happening?

  1. 1.Natural numbers increase by 1.
  2. 2.Odd numbers increase by 2.
  3. 3.Triangular numbers are formed by adding 1, then 2, then 3, then 4, and so on.
  4. 4.Square numbers are 1²,2²,3²,4²,… .
  5. 5.Different sequences can follow very different rules.

The three dots … mean that the pattern continues. Some sequences continue forever, while others stop after a fixed number of terms.

Definition
Finite Sequence

A sequence with a fixed number of terms.

Definition
Infinite Sequence

A sequence that continues without end.

Daily-Life Example: Weekly Savings

Problem
Suppose you save ₹50 in Week 1, ₹100 in Week 2, ₹150 in Week 3 and ₹200 in Week 4. What sequence do you see?

  1. 1.The sequence is 50,100,150,200,… .
  2. 2.Each term is ₹50 more than the previous term.
  3. 3.The order matters because each amount is linked to a week number.

Triangular Numbers

The sequence 1,3,6,10,15,… is special because each term is the sum of natural numbers up to that position. The fifth triangular number is 1+2+3+4+5=15.

Triangular numbers as dot patterns The first five triangular numbers shown as triangular dot arrays. Each new stage adds one new row of dots. Triangular Numbers as Dot Patterns Each stage is formed by adding one new row Stage 1 1 1 row Stage 2 3 1 + 2 Stage 3 6 1 + 2 + 3 Stage 4 10 1 + 2 + 3 + 4 Stage 5 15 1 + 2 + 3 + 4 + 5 Existing dots New row added Tₙ = 1 + 2 + 3 + ⋯ + n = n(n + 1) ÷ 2
Triangular numbers as dot patterns

Square Numbers and Odd Numbers

Square numbers also reveal a hidden pattern: 1,4,9,16,25,… . The differences are 3,5,7,9,… . In other words, each square can be built from the previous square by adding the next odd number.

Square-number patternLaTeX
Square numbers growing by odd-number layers Square dot arrays from one by one to four by four. Each new square is formed by adding an L-shaped layer containing 3, 5, or 7 dots. Square Numbers Growing by Odd-Number Layers Every larger square is made by adding an L-shaped layer of dots +3 +5 +7 1 × 1 square 1 = 1² Starting square 2 × 2 square 4 = 2² New L-layer: 3 dots 3 × 3 square 9 = 3² New L-layer: 5 dots 4 × 4 square 16 = 4² New L-layer: 7 dots Existing square New odd-number layer 1 → 1 + 3 = 4 → 4 + 5 = 9 → 9 + 7 = 16
Square numbers growing by odd-number layers

Term Notation

To talk precisely about positions, we write t₁ for the first term, t₂ for the second term, and tₙ for the term in position n. The subscript tells us the position; it is not multiplication.

Reading Term Notation

Problem
For the odd-number sequence 1,3,5,7,9,…, what do t₁, t₄ and t₅ mean?

  1. 1.t₁=1 because 1 is in the first position.
  2. 2.t₄=7 because 7 is in the fourth position.
  3. 3.t₅=9 because 9 is in the fifth position.

Practice Problems

Practice Problems
  1. Write the next four terms of 2,5,8,11,14,… .
  2. For the sequence 1,4,9,16,25,… write t₂, t₄ and t₆.
  3. State whether each is finite or infinite: months in one year; natural numbers; scores from five matches.
  4. Write the first six triangular numbers.
  5. Explain why the order of terms matters in a sequence.
  6. Create one sequence from a daily-life situation and describe its rule.

Key Takeaways

Key Takeaways

• A sequence is an ordered list. • Each entry is called a term. • Sequences may be finite or infinite. • Familiar patterns include odd, triangular and square numbers. • tₙ means the term in position n.

Coming Next

Next, we learn how one formula can directly tell us any term of a sequence—even the 100th or 1000th term.

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Next · Lesson 2

Explicit Rules for Sequences