1st Year Math Chapter 6 Exercise 6.5 Notes: Word Problems on 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 6.5 applies the arithmetic progression concepts from Exercises 6.2 through 6.4 to real-world word problems, where the first step is recognizing that a situation follows an A.P. pattern.
What Does This Exercise Cover?
This exercise builds the practical skill of translating a described situation — savings, seating, distances — into the language of an A.P., then solving for a specific term or a total sum.
Key Concepts
1. Translating a Word Problem into an A.P.
The key first step is identifying the first term (the starting value) and the common difference (the fixed amount of change per step) from the problem’s description.
2. Choosing Between a Term and a Sum
If the question asks for a value at a specific point (like ‘the 8th month’), use the nth term formula. If it asks for a total across several steps, use the sum formula from Exercise 6.4.
3. Common A.P. Scenarios
Typical situations that model an A.P. include savings that increase by a fixed amount, seating arrangements with a fixed increase per row, and distances that change by a constant amount over time.
Solved Examples
Example 1: A man saves Rs. 200 in the first month and increases his savings by Rs. 50 every following month. How much does he save in the 8th month?
a₁ = 200, d = 50, n = 8.
a₈ = 200 + 7(50) = 200 + 350.
Answer: Rs. 550
Example 2: A theater’s first row has 20 seats, and each row after has 3 more seats than the one before. Find the number of seats in the 12th row, and the total seats in the first 12 rows.
a₁ = 20, d = 3, n = 12.
a₁₂ = 20 + 11(3) = 20 + 33 = 53.
S₁₂ = (12/2)(20 + 53) = 6(73).
Answer: 12th row: 53 seats; total: 438 seats
Sample MCQs
1. A situation involving equal fixed increases typically models:
a) A geometric progression b) An arithmetic progression c) A harmonic progression d) None of these
Answer: b) An arithmetic progression
2. If savings increase by a fixed amount each month, the first term represents:
a) The total savings b) The savings in the first month c) The common difference d) The number of months
Answer: b) The savings in the first month
3. To find the total amount saved over several months in an A.P. scenario, use:
a) The nth term formula only b) The sum formula c) The arithmetic mean formula d) None of these
Answer: b) The sum formula
4. A theater with rows increasing by a fixed number of seats each row models:
a) A geometric series b) An arithmetic series c) An infinite series d) A harmonic series
Answer: b) An arithmetic series
5. In word problems, ‘common difference’ most often represents:
a) The total b) The fixed amount of increase or decrease c) The first term d) The last term
Answer: b) The fixed amount of increase or decrease
Important Short Questions
- Explain how to identify the first term and common difference in an A.P. word problem.
- A worker’s salary increases by a fixed amount each year. What kind of sequence does this describe?
- If a₁ = 100 and d = 25, find the 6th term in a savings scenario.
- Which formula would you use to find the total across several terms in an A.P. word problem?
- Explain the difference between finding ‘a specific term’ and finding ‘the total’ in a word problem.
Important Long Questions
- A man saves Rs. 500 in the first month and increases his savings by Rs. 100 each month. Find his savings in the 10th month and his total savings over the first 10 months.
- A stadium has 25 seats in the first row, with each subsequent row having 4 more seats than the one before it. Find the number of seats in the 15th row and the total number of seats in the first 15 rows.
- A ball travels 10 meters in the first second after being dropped, and 2 meters less in each subsequent second. Find how far it travels in the 5th second, and the total distance covered over 5 seconds.
- Explain, using a real-world example of your own, how a situation can be modeled as an arithmetic progression.
How to Approach This Exercise Effectively
- Underline the starting value and the fixed change described in the problem before writing any formula — this prevents mixing up a₁ and d.
- Decide immediately whether the question wants a specific term or a running total, since this determines which formula to reach for.
- Write out the first two or three terms by hand from the word problem as a sanity check before applying a formula.
- For multi-part questions (term and total), solve the term first — you’ll often need it (as aₙ) for the sum formula.
- Practice translating a few real scenarios of your own into A.P. language to build confidence beyond the standard textbook wording.
FAQs
Q: How do I know if a word problem is describing an A.P.?
A: Look for a fixed amount being added or subtracted at each step — a constant rate of change is the hallmark of an arithmetic progression.
Q: What’s the difference between asking for a ‘term’ and asking for a ‘sum’?
A: A term question wants one specific value at one point (like the 8th month), while a sum question wants the combined total of several terms added together.
Q: Do these techniques apply outside finance and seating examples?
A: Yes — any situation with a constant rate of increase or decrease, such as distances, temperatures, or production counts, can be modeled the same way.
Q: What if the common difference in a word problem is negative?
A: The same formulas still apply directly — a negative d simply means the described quantity is decreasing rather than increasing over time.
Notes prepared for Punjab Board (PECTAA) 1st Year Mathematics, 2026-27 session, Chapter 6: Sequences and Series, Exercise 6.5.