1st Year Math Chapter 6 Exercise 6.9 Notes: Word Problems on 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 6.9 applies the geometric progression concepts from Exercises 6.6 through 6.8 to real-world word problems, where a quantity grows or shrinks by a fixed percentage or factor at each step.
What Does This Exercise Cover?
This exercise builds the practical skill of translating growth-and-decay situations — population, investment, bouncing balls — into G.P. language, then solving for a specific term or a total.
Key Concepts
1. Translating a Word Problem into a G.P.
The key step is identifying the starting value (first term) and the fixed multiplying factor per period (common ratio) described in the problem.
2. Growth vs Decay Scenarios
If the common ratio is greater than 1, the quantity grows over time (as in population growth or compound interest). If the ratio is between 0 and 1, the quantity shrinks over time (as in depreciation or a bouncing ball losing height).
3. Common G.P. Scenarios
Typical situations include population doubling or tripling, compound interest, radioactive decay, and objects losing a fixed percentage of height or value repeatedly.
Solved Examples
Example 1: A population of bacteria starts at 100 and doubles every hour. Find the population after 5 hours.
a₁ = 100, r = 2.
Population after 5 hours corresponds to the 6th term (starting count plus 5 doublings): a₆ = 100 × 2⁵ = 100 × 32.
Answer: 3200 bacteria
Example 2: An investment of Rs. 10,000 grows at a compound interest rate of 10% annually. Find its value after 3 years.
a₁ = 10,000, r = 1.10 (adding 10% each year).
Value after 3 years: a₁ × r³ = 10,000 × (1.10)³ = 10,000 × 1.331.
Answer: Rs. 13,310
Sample MCQs
1. A situation where a value multiplies by a fixed factor each period models:
a) An arithmetic progression b) A geometric progression c) A harmonic progression d) None of these
Answer: b) A geometric progression
2. In a population-doubling problem, the common ratio is:
a) 1 b) 2 c) 0.5 d) 0
Answer: b) 2
3. Compound interest problems are typically modeled using:
a) An A.P. b) A G.P. c) An H.P. d) A linear equation
Answer: b) A G.P.
4. If a quantity decreases by a fixed percentage each period, the common ratio is:
a) Greater than 1 b) Between 0 and 1 c) Negative d) Exactly 1
Answer: b) Between 0 and 1
5. To find the value after several multiplying periods, use:
a) The nth term of a G.P. b) The arithmetic mean c) The common difference d) None of these
Answer: a) The nth term of a G.P.
Important Short Questions
- Explain how to identify the first term and common ratio in a G.P. word problem.
- A car’s value depreciates by a fixed percentage each year. What kind of sequence does this describe?
- If a population starts at 500 and doubles every year, find the population after 3 years.
- What does the common ratio represent in a compound interest scenario?
- Explain the difference between a growth scenario (r > 1) and a decay scenario (0 < r < 1) in a G.P.
Important Long Questions
- A culture of bacteria starts with 200 cells and triples every hour. Find the number of cells after 4 hours.
- An investment of Rs. 10,000 grows at a compound interest rate of 10% annually. Find its value after 3 years.
- A ball dropped from 30 m bounces back to 50% of its previous height each time. Find the height it reaches after the 3rd bounce.
- Explain, using a real-world example of your own, how a situation can be modeled as a geometric progression.
How to Approach This Exercise Effectively
- Identify whether the situation is growing or shrinking first — this tells you immediately whether to expect r > 1 or 0 < r < 1.
- For percentage-based problems, convert the percentage into a ratio carefully: a 10% increase means r = 1.10, not r = 0.10.
- Write out the first two or three terms by hand from the word problem before jumping to the general formula, to make sure your setup is correct.
- For bouncing-ball or repeated-halving problems, be careful about whether the ‘first term’ is the original height or the height after the first bounce.
- Practice a mix of growth and decay word problems, since exams often test both in the same paper.
FAQs
Q: How do I know if a word problem describes a G.P. rather than an A.P.?
A: Look for a fixed multiplying factor or percentage change at each step, rather than a fixed amount being added or subtracted.
Q: What’s the difference between growth and decay in these problems?
A: Growth scenarios have a common ratio greater than 1 (the quantity increases), while decay scenarios have a ratio between 0 and 1 (the quantity decreases).
Q: Can the common ratio in a real-world G.P. be negative?
A: It’s uncommon in typical growth/decay word problems, since most real quantities like population or money don’t alternate in sign, but mathematically a negative ratio is still valid.
Q: How is compound interest related to geometric progressions?
A: Each year’s balance is the previous year’s balance multiplied by a fixed factor (1 + interest rate), which is exactly the defining property of a G.P.
Notes prepared for Punjab Board (PECTAA) 1st Year Mathematics, 2026-27 session, Chapter 6: Sequences and Series, Exercise 6.9.