1st Year Math Chapter 6 Exercise 6.2 Notes: Arithmetic Progression (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 6.2 introduces the Arithmetic Progression (A.P.) — the first specific type of sequence in the chapter — where each term increases or decreases from the last by the same fixed amount.
What Does This Exercise Cover?
This exercise teaches how to recognize an A.P., identify its common difference, and use the general term formula to find any term of the progression.
Key Concepts
1. What Is an Arithmetic Progression?
A sequence is an arithmetic progression if the difference between any two consecutive terms is always the same. This constant difference is called the common difference, denoted d.
2. The General (nth) Term of an A.P.
If the first term is a₁ and the common difference is d, the nth term is given by aₙ = a₁ + (n − 1)d.
3. Finding the Common Difference
The common difference can be found by subtracting any term from the term immediately after it: d = aₙ − aₙ₋₁.
4. Recognizing an A.P.
To check whether a sequence is an A.P., subtract each term from the one after it — if every result is the same, the sequence is an A.P.
Solved Examples
Example 1: Find the 10th term of the A.P. 3, 7, 11, 15, …
a₁ = 3, d = 4.
a₁₀ = 3 + (10 − 1)(4) = 3 + 36.
Answer: a₁₀ = 39
Example 2: Determine whether 5, 8, 12, 15 is an A.P.
Differences: 8 − 5 = 3, 12 − 8 = 4, 15 − 12 = 3.
The differences are not all equal.
Answer: Not an A.P.
Example 3: If the 5th term of an A.P. is 17 and the common difference is 3, find the first term.
a₅ = a₁ + 4d, so 17 = a₁ + 4(3) = a₁ + 12.
Answer: a₁ = 5
Sample MCQs
1. In an A.P., the difference between any two consecutive terms is:
a) Always increasing b) Always constant c) Always zero d) Always negative
Answer: b) Always constant
2. The general term of an A.P. is given by:
a) aₙ = a₁ × d^(n−1) b) aₙ = a₁ + (n − 1)d c) aₙ = a₁ − nd d) aₙ = a₁ / n
Answer: b) aₙ = a₁ + (n − 1)d
3. The common difference of 4, 9, 14, 19, … is:
a) 4 b) 5 c) 9 d) 14
Answer: b) 5
4. If a₁ = 2 and d = 3, the 6th term is:
a) 15 b) 17 c) 18 d) 20
Answer: b) 17
5. The sequence 2, 4, 8, 16, … is:
a) An A.P. b) Not an A.P. c) Constant d) Undefined
Answer: b) Not an A.P.
Important Short Questions
- Define an arithmetic progression.
- State the formula for the nth term of an A.P.
- Find the common difference of the A.P. 10, 7, 4, 1, …
- Determine whether 3, 6, 10, 15 is an A.P.
- Find the first term of an A.P. whose 4th term is 20 and common difference is 4.
Important Long Questions
- Find the 15th term of the A.P. 5, 9, 13, 17, …
- If the 3rd term of an A.P. is 8 and the 7th term is 20, find the first term and common difference.
- Determine which term of the A.P. 2, 7, 12, 17, … equals 97.
- Explain how to test whether a given sequence is an A.P., using 4, 7, 10, 13 as an example.
How to Approach This Exercise Effectively
- Always calculate the common difference first — every other calculation in this exercise depends on it.
- When two terms are given instead of the first term and d, set up two equations using aₙ = a₁ + (n−1)d and solve them together.
- To find ‘which term equals a given value,’ set aₙ equal to that value and solve for n.
- Watch for negative common differences — a decreasing A.P. follows exactly the same formula.
- Double-check your answer for a specific term by working out a couple of terms by hand and comparing the pattern.
FAQs
Q: What does ‘common difference’ mean?
A: It’s the fixed amount added (or subtracted) to get from one term of an A.P. to the next.
Q: Can the common difference be negative?
A: Yes — a negative common difference simply means the sequence is decreasing rather than increasing.
Q: How can I tell quickly if a sequence is an A.P.?
A: Subtract each pair of consecutive terms; if every difference comes out the same, the sequence is an A.P.
Q: What if I’m given two terms but not the first term?
A: Write both terms using the general formula aₙ = a₁ + (n−1)d, then solve the resulting pair of equations simultaneously for a₁ and d.
Notes prepared for Punjab Board (PECTAA) 1st Year Mathematics, 2026-27 session, Chapter 6: Sequences and Series, Exercise 6.2.