1st Year Math Chapter 6 Exercise 6.6 Notes: Geometric 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 6.6 introduces the Geometric Progression (G.P.), a second major type of sequence, where each term is found by multiplying the previous term by a fixed ratio rather than adding a fixed amount.
What Does This Exercise Cover?
This exercise teaches how to recognize a G.P., identify its common ratio, and use the general term formula to find any term of the progression.
Key Concepts
1. What Is a Geometric Progression?
A sequence is a geometric progression if the ratio between any two consecutive terms is always the same. This constant ratio is called the common ratio, denoted r.
2. The General (nth) Term of a G.P.
If the first term is a₁ and the common ratio is r, the nth term is given by aₙ = a₁ × r^(n−1).
3. Finding the Common Ratio
The common ratio can be found by dividing any term by the term immediately before it: r = aₙ / aₙ₋₁.
4. Recognizing a G.P.
To check whether a sequence is a G.P., divide each term by the one before it — if every result is the same, the sequence is a G.P.
Solved Examples
Example 1: Find the 6th term of the G.P. 2, 6, 18, 54, …
a₁ = 2, r = 3.
a₆ = 2 × 3⁵ = 2 × 243.
Answer: a₆ = 486
Example 2: Determine whether 5, 10, 20, 40 is a G.P.
Ratios: 10/5 = 2, 20/10 = 2, 40/20 = 2.
All ratios are equal.
Answer: Yes, it is a G.P. with r = 2
Example 3: If the first term of a G.P. is 3 and the common ratio is 1/2, find the 5th term.
a₅ = 3 × (1/2)⁴ = 3 × 1/16.
Answer: a₅ = 3/16
Sample MCQs
1. In a G.P., the ratio between consecutive terms is:
a) Always increasing b) Always constant c) Always 1 d) Always zero
Answer: b) Always constant
2. The general term of a G.P. is given by:
a) aₙ = a₁ + (n−1)r b) aₙ = a₁ × r^(n−1) c) aₙ = a₁ − r d) aₙ = a₁ / r^n
Answer: b) aₙ = a₁ × r^(n−1)
3. The common ratio of 4, 12, 36, 108, … is:
a) 3 b) 4 c) 8 d) 12
Answer: a) 3
4. If a₁ = 5 and r = 2, the 4th term is:
a) 20 b) 30 c) 40 d) 50
Answer: c) 40
5. The sequence 3, 6, 9, 12, … is:
a) A G.P. b) Not a G.P. c) Constant d) Undefined
Answer: b) Not a G.P.
Important Short Questions
- Define a geometric progression.
- State the formula for the nth term of a G.P.
- Find the common ratio of the G.P. 81, 27, 9, 3, …
- Determine whether 2, 4, 6, 8 is a G.P.
- Find the 4th term of a G.P. whose first term is 2 and common ratio is 3.
Important Long Questions
- Find the 8th term of the G.P. 1, 3, 9, 27, …
- If the 2nd term of a G.P. is 6 and the 5th term is 162, find the first term and common ratio.
- Determine which term of the G.P. 4, 8, 16, 32, … equals 1024.
- Explain how to test whether a given sequence is a G.P., using 2, 6, 18, 54 as an example.
How to Approach This Exercise Effectively
- Always compute the common ratio first by dividing consecutive terms — every other step in this exercise depends on it.
- When two non-consecutive terms are given, divide one general-term equation by the other to eliminate a₁ and solve for r directly.
- Watch for negative common ratios, which make a G.P. alternate in sign from term to term.
- Distinguish a G.P. from an A.P. quickly: an A.P. has a constant difference, a G.P. has a constant ratio — check both if unsure.
- Practice with fractional common ratios (like 1/2 or 1/3), since these appear frequently and are easy to miscalculate.
FAQs
Q: What does ‘common ratio’ mean?
A: It’s the fixed number you multiply one term by to get the next term in a geometric progression.
Q: Can the common ratio be negative or a fraction?
A: Yes — a negative ratio makes terms alternate in sign, and a fractional ratio (between −1 and 1) makes the terms shrink toward zero.
Q: How can I quickly tell a G.P. apart from an A.P.?
A: Check consecutive terms: constant differences mean an A.P., while constant ratios mean a G.P.
Q: What happens to a G.P. if the common ratio is between −1 and 1?
A: The terms get smaller and smaller in size, approaching zero as n increases — this behavior becomes important later, in Exercise 6.8’s infinite series.
Notes prepared for Punjab Board (PECTAA) 1st Year Mathematics, 2026-27 session, Chapter 6: Sequences and Series, Exercise 6.6.