1st Year Math Chapter 6 Exercise 6.1 Notes Sequences — Definition and Types (Punjab Board 2026-27)
Complete concept notes, formulas, and solved examples for ICS/FSc Part 1 Mathematics, Punjab Board (PECTAA), Single National Curriculum 2026-27 session.Chapter 6, Sequences and Series, follows Partial Fractions in the 14-unit Mathematics 11 (PECTAA) textbook. Exercise 6.1 opens the chapter by defining what a sequence is and introducing the general term that generates it.
What Does This Exercise Cover?
This exercise builds the vocabulary needed for the rest of the chapter — sequences, terms, finite and infinite sequences, and the general (nth) term formula — before arithmetic and geometric progressions are introduced.
Key Concepts
1. What Is a Sequence?
A sequence is an ordered list of numbers, called terms, arranged according to a specific rule. Formally, it is a function whose domain is the set of natural numbers, with each output written as a1, a2, a3, and so on.
2. Finite and Infinite Sequences
A finite sequence has a limited, countable number of terms. An infinite sequence continues without end, following its rule indefinitely.
3. The General (nth) Term
The general term, written aₙ, is a formula in terms of n that produces any term of the sequence when a specific value of n is substituted.
4. Explicit vs Recursive Definitions
A sequence is defined explicitly when aₙ is given directly as a formula in n. It is defined recursively when each term is calculated from the term(s) before it, along with a starting value.
Solved Examples
Example 1: Write the first four terms of the sequence with nth term aₙ = 2n + 3.
Substitute n = 1, 2, 3, 4 in turn: a₁ = 2(1)+3, a₂ = 2(2)+3, a₃ = 2(3)+3, a₄ = 2(4)+3.
Answer: 5, 7, 9, 11
Example 2: Find the general term of the sequence 3, 6, 9, 12, …
Each term is 3 times its position number n.
Answer: aₙ = 3n
Example 3: A sequence is defined recursively by a₁ = 2, aₙ = aₙ₋₁ + 5. Find the first four terms.
a₁ = 2.
a₂ = a₁ + 5 = 7.
a₃ = a₂ + 5 = 12.
a₄ = a₃ + 5 = 17.
Answer: 2, 7, 12, 17
Sample MCQs
1. A sequence is best described as:
a) An unordered set of numbers b) An ordered list of numbers following a rule c) A single number d) An equation with two variables
Answer: b) An ordered list of numbers following a rule
2. If aₙ = 3n − 1, then a₅ equals:
a) 13 b) 14 c) 15 d) 16
Answer: b) 14
3. A sequence with a limited number of terms is called:
a) Infinite b) Finite c) Recursive d) Arithmetic
Answer: b) Finite
4. In a recursively defined sequence, each term is found using:
a) Only the position number n b) The previous term(s) c) A random number d) Only the last term of the sequence
Answer: b) The previous term(s)
5. The ‘general term’ of a sequence is also known as its:
a) First term b) nth term c) Common difference d) Sum
Answer: b) nth term
Important Short Questions
- Define a sequence.
- What is the difference between a finite and an infinite sequence?
- What is meant by the general (nth) term of a sequence?
- Differentiate between an explicit and a recursive definition of a sequence.
- Find the first three terms of the sequence with nth term aₙ = n² − 1.
Important Long Questions
- Write the first five terms of the sequence aₙ = 2n² − n + 1, and describe the pattern in the differences between consecutive terms.
- A sequence is defined recursively as a₁ = 3, aₙ = 2aₙ₋₁ − 1. Find the first five terms.
- Find the general term of the sequence 2, 5, 8, 11, … and use it to find the 10th term.
- Explain the difference between a sequence and a series, giving one example of each.
How to Approach This Exercise Effectively
- Always test a proposed general term formula against at least the first two or three known terms before trusting it.
- When working with a recursive definition, write out each term in order — skipping ahead usually leads to mistakes.
- Get comfortable moving between explicit and recursive descriptions of the same sequence; both appear on exams.
- Keep ‘sequence’ and ‘series’ mentally separate from the start of this chapter — the two terms are often confused.
- Practice spotting simple patterns (like multiplying by n, or adding a fixed amount) before assuming a sequence needs a complicated formula.
FAQs
Q: What is the difference between a sequence and a series?
A: A sequence is an ordered list of terms, while a series is the sum of those terms — series are covered in more depth in Exercise 6.4.
Q: What does ‘general term’ mean?
A: It’s a single formula, written in terms of n, that generates any term of the sequence by substituting the desired position number.
Q: Can a sequence’s terms decrease instead of increase?
A: Yes — a sequence can increase, decrease, alternate in sign, or follow any consistent rule; there’s no requirement that terms only grow.
Q: Is every sequence either arithmetic or geometric?
A: No — many sequences follow other rules entirely; arithmetic and geometric progressions are just two common, well-studied types covered later in this chapter.
Notes prepared for Punjab Board (PECTAA) 1st Year Mathematics, 2026-27 session, Chapter 6: Sequences and Series, Exercise 6.1.