1st Year Math Chapter 6 Exercise 6.3 Notes: Arithmetic Mean and Arithmetic Means (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 6.3 builds on the A.P. concepts from Exercise 6.2, focusing on the arithmetic mean between two numbers, and how to insert several evenly-spaced arithmetic means between them.
What Does This Exercise Cover?
This exercise teaches how to find a single number that fits exactly between two others in an A.P. sense, and how to extend that idea to inserting several such numbers at once.
Key Concepts
1. Arithmetic Mean Between Two Numbers
The arithmetic mean (A.M.) between two numbers a and b is the number A such that a, A, b form an A.P. This gives the familiar formula A = (a + b) / 2.
2. n Arithmetic Means Between Two Numbers
To insert n arithmetic means between a and b means finding n numbers so that, together with a and b, all n + 2 terms form a single A.P.
3. Finding the Common Difference for n Means
Since a is the first term and b is the (n+2)th term of the resulting A.P., the common difference is d = (b − a) / (n + 1).
Solved Examples
Example 1: Find the arithmetic mean between 8 and 14.
A = (8 + 14) / 2.
Answer: A = 11
Example 2: Insert 3 arithmetic means between 2 and 18.
d = (18 − 2) / (3 + 1) = 16/4 = 4.
Means: 2 + 4 = 6, 6 + 4 = 10, 10 + 4 = 14.
Answer: 6, 10, 14 (full sequence: 2, 6, 10, 14, 18)
Sample MCQs
1. The arithmetic mean of two numbers a and b is:
a) a + b b) (a + b) / 2 c) ab d) a − b
Answer: b) (a + b) / 2
2. To insert n arithmetic means between a and b, the common difference is:
a) (b − a) / n b) (b − a) / (n + 1) c) (b − a) / (n − 1) d) (b + a) / n
Answer: b) (b − a) / (n + 1)
3. The arithmetic mean between 5 and 15 is:
a) 8 b) 10 c) 12 d) 20
Answer: b) 10
4. Inserting 2 arithmetic means between 4 and 16 gives a common difference of:
a) 3 b) 4 c) 6 d) 12
Answer: b) 4
5. If A is the arithmetic mean between a and b, then a, A, b form:
a) A G.P. b) An A.P. c) An H.P. d) None of these
Answer: b) An A.P.
Important Short Questions
- Define the arithmetic mean between two numbers.
- State the formula for finding the common difference when inserting n arithmetic means between two numbers.
- Find the arithmetic mean between 20 and 30.
- How many terms are there in total after inserting 4 arithmetic means between two given numbers?
- Insert one arithmetic mean between 7 and 21.
Important Long Questions
- Insert 4 arithmetic means between 3 and 23, listing the complete sequence.
- Find the arithmetic mean between (2x + 1) and (4x − 3), simplifying your answer.
- If 5 arithmetic means are inserted between 10 and 40, find the third mean.
- Explain why inserting n arithmetic means between a and b always results in an A.P. of n + 2 terms.
How to Approach This Exercise Effectively
- Always count carefully: inserting n means between two numbers creates a sequence of n + 2 total terms, including the two originals.
- Compute the common difference once, then generate every mean by repeatedly adding it — don’t recompute d for each mean.
- For algebraic means (like between 2x+1 and 4x−3), simplify fully and double-check by substituting a sample value of x.
- Remember ‘the arithmetic mean’ (singular) always refers to just one number, exactly halfway between two values.
- Practice both numeric and algebraic examples, since exam questions mix both styles.
FAQs
Q: What is the difference between ‘the arithmetic mean’ and ‘arithmetic means’ (plural)?
A: ‘The arithmetic mean’ refers to the single midpoint value between two numbers, while ‘arithmetic means’ (plural) refers to several evenly-spaced values inserted between them.
Q: Why is the formula (b − a)/(n + 1) used instead of dividing by n?
A: Because inserting n means creates n + 1 equal gaps between the first term a and the last term b, not n gaps.
Q: Can arithmetic means be negative or fractional?
A: Yes — arithmetic means can be negative, fractional, or even algebraic expressions, depending on the values of a and b.
Q: Is the arithmetic mean always exactly halfway between two numbers?
A: Yes, by definition — the arithmetic mean is always the midpoint value on the number line between the two given numbers.
Notes prepared for Punjab Board (PECTAA) 1st Year Mathematics, 2026-27 session, Chapter 6: Sequences and Series, Exercise 6.3.