1st Year Math Chapter 6 Exercise 6.4 Notes: Series — Sum of an Arithmetic Series (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.
Exercise 6.4 introduces the idea of a series — the sum of a sequence’s terms — and derives the formulas used to add up the terms of an arithmetic progression.
What Does This Exercise Cover?
This exercise shifts from finding individual terms of an A.P. to finding the total of many terms added together, using two closely related sum formulas.
Key Concepts
1. What Is a Series?
A series is the sum of the terms of a sequence. If a sequence has terms a₁, a₂, a₃, …, aₙ, the corresponding series is a₁ + a₂ + a₃ + … + aₙ.
2. Sum of n Terms of an A.P.
The sum of the first n terms of an A.P., denoted Sₙ, can be found using Sₙ = (n/2)[2a₁ + (n − 1)d], or equivalently Sₙ = (n/2)(a₁ + aₙ) when the last term is already known.
Sum Formulas for an Arithmetic Series
| Formula | Best Used When |
| Sₙ = (n/2)[2a₁ + (n − 1)d] | You know the first term and common difference |
| Sₙ = (n/2)(a₁ + aₙ) | You already know (or can quickly find) the last term |
Solved Examples
Example 1: Find the sum of the first 10 terms of the A.P. 3, 7, 11, …
a₁ = 3, d = 4, n = 10.
Sₙ = (10/2)[2(3) + 9(4)] = 5[6 + 36] = 5(42).
Answer: S₁₀ = 210
Example 2: Find the sum of the A.P. 2 + 5 + 8 + … + 50.
a₁ = 2, d = 3, aₙ = 50.
Find n: 50 = 2 + (n − 1)(3), so 48 = 3(n − 1), giving n = 17.
Sₙ = (17/2)(2 + 50) = (17/2)(52).
Answer: S = 442
Sample MCQs
1. A series is:
a) An ordered list of numbers b) The sum of the terms of a sequence c) A type of function d) A single number
Answer: b) The sum of the terms of a sequence
2. The formula for the sum of n terms of an A.P. is:
a) Sₙ = n × a₁ × d b) Sₙ = (n/2)[2a₁ + (n−1)d] c) Sₙ = a₁ + (n−1)d d) Sₙ = n × d
Answer: b) Sₙ = (n/2)[2a₁ + (n−1)d]
3. The sum of the first 10 terms of 3, 7, 11, … is:
a) 39 b) 175 c) 210 d) 220
Answer: c) 210
4. Sₙ = (n/2)(a₁ + aₙ) is most convenient when:
a) d is unknown b) The last term aₙ is known c) n is unknown d) a₁ is unknown
Answer: b) The last term aₙ is known
5. The sum of an A.P. depends on:
a) Only the first term b) Only the common difference c) The first term, common difference, and number of terms d) Only the number of terms
Answer: c) The first term, common difference, and number of terms
Important Short Questions
- Define a series and explain how it differs from a sequence.
- State the two common formulas for the sum of n terms of an A.P.
- Find the sum of the first 8 terms of the A.P. 5, 9, 13, …
- When is it easier to use Sₙ = (n/2)(a₁ + aₙ) instead of Sₙ = (n/2)[2a₁ + (n−1)d]?
- Find the sum of the first 20 natural numbers using the A.P. sum formula.
Important Long Questions
- Find the sum of the A.P. 2 + 5 + 8 + … + 50, showing how you find the number of terms first.
- Find the sum of the first 15 terms of an A.P. whose first term is 4 and common difference is 3.
- The sum of the first n terms of an A.P. is 120, the first term is 3, and the common difference is 2. Find n.
- Explain conceptually why Sₙ = (n/2)[2a₁ + (n−1)d], using the idea of pairing terms from the start and end of the series.
How to Approach This Exercise Effectively
- Keep ‘sequence’ (the list of terms) and ‘series’ (their sum) clearly distinct in your own explanations — examiners often test this directly.
- If a problem gives you the last term, use Sₙ = (n/2)(a₁+aₙ) to skip an extra step.
- When you only know a1, d, and need n from a target last term, solve for n first before attempting the sum.
- For ‘find n’ problems, set your sum formula equal to the given total and solve the resulting equation for n carefully — it’s often quadratic.
- Practice deriving the sum formula once by pairing the first and last terms — understanding where it comes from makes it far easier to recall under exam pressure.
FAQs
Q: What’s the difference between a sequence and a series?
A: A sequence is the ordered list of individual terms; a series is what you get when you add all of those terms together.
Q: Which sum formula should I use first?
A: Use Sₙ = (n/2)(a₁+aₙ) if the last term is already known or easy to find; otherwise use Sₙ = (n/2)[2a₁+(n−1)d].
Q: Can the sum of an A.P. ever be negative?
A: Yes — if the terms themselves are mostly negative (for example, a decreasing A.P. that goes well below zero), their sum can be negative too.
Q: Does the sum formula work for a decreasing A.P.?
A: Yes — both formulas work identically whether the common difference d is positive or negative.
Notes prepared for Punjab Board (PECTAA) 1st Year Mathematics, 2026-27 session, Chapter 6: Sequences and Series, Exercise 6.4.