1st Year Math Chapter 6 Exercise 6.7 Notes: Geometric Means and n Geometric 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 6.7 mirrors Exercise 6.3, but for geometric progressions — introducing the geometric mean between two numbers, and how to insert several geometric means between them.
What Does This Exercise Cover?
This exercise teaches how to find a single number that fits exactly between two others in a G.P. sense, and how to extend that idea to inserting several such numbers at once.
Key Concepts
1. Geometric Mean Between Two Numbers
The geometric mean (G.M.) between two positive numbers a and b is the positive number G such that a, G, b form a G.P. This gives the formula G = √(ab).
2. n Geometric Means Between Two Numbers
To insert n geometric means between a and b means finding n numbers so that, together with a and b, all n + 2 terms form a single G.P.
3. Finding the Common Ratio for n Means
Since a is the first term and b is the (n+2)th term of the resulting G.P., the common ratio is r = (b/a)^(1/(n+1)).
Solved Examples
Example 1: Find the geometric mean between 4 and 16.
G = √(4 × 16) = √64.
Answer: G = 8
Example 2: Insert 2 geometric means between 2 and 54.
r = (54/2)^(1/3) = 27^(1/3) = 3.
Means: 2 × 3 = 6, 6 × 3 = 18.
Answer: 6, 18 (full sequence: 2, 6, 18, 54)
Sample MCQs
1. The geometric mean of two positive numbers a and b is:
a) (a + b) / 2 b) √(ab) c) ab d) a − b
Answer: b) √(ab)
2. To insert n geometric means between a and b, the common ratio is:
a) (b/a)^n b) (b/a)^(1/(n+1)) c) (b − a)/(n + 1) d) (b/a)^(1/n)
Answer: b) (b/a)^(1/(n+1))
3. The geometric mean between 9 and 25 is:
a) 15 b) 16 c) 17 d) 20
Answer: a) 15
4. Inserting 1 geometric mean between 4 and 36 gives:
a) 12 b) 16 c) 20 d) 24
Answer: a) 12
5. If G is the geometric mean between a and b, then a, G, b form:
a) An A.P. b) A G.P. c) An H.P. d) None of these
Answer: b) A G.P.
Important Short Questions
- Define the geometric mean between two numbers.
- State the formula for finding the common ratio when inserting n geometric means between two numbers.
- Find the geometric mean between 5 and 45.
- How many terms are there in total after inserting 3 geometric means between two given numbers?
- Insert one geometric mean between 8 and 32.
Important Long Questions
- Insert 3 geometric means between 2 and 32, listing the complete sequence.
- Find the geometric mean between 6 and 24, and verify that the resulting three-term sequence is a G.P.
- If 2 geometric means are inserted between 5 and 135, find both means.
- Explain why the geometric mean formula √(ab) only applies to positive numbers a and b.
How to Approach This Exercise Effectively
- Compute a × b first, then take the square root — this two-step process avoids arithmetic slips in geometric mean questions.
- For n geometric means, calculate r once, then generate each mean by repeatedly multiplying, just as with arithmetic means and addition.
- Remember the count: inserting n geometric means between two numbers produces a total of n + 2 terms, including the two originals.
- Compare this exercise side-by-side with Exercise 6.3 — the structure is identical, just addition/subtraction replaced with multiplication/roots.
- Double check your final sequence by confirming the ratio between every consecutive pair is the same.
FAQs
Q: What is the difference between ‘the geometric mean’ and ‘geometric means’ (plural)?
A: ‘The geometric mean’ is the single value exactly between two numbers in a G.P. sense, while ‘geometric means’ (plural) are several such values inserted together.
Q: Why must a and b be positive for a real geometric mean to exist?
A: Because the formula involves a square root of the product ab; if ab were negative, the square root would not be a real number.
Q: How does this relate to inserting arithmetic means?
A: The logic is identical, but arithmetic means use addition and division for the common difference, while geometric means use multiplication and roots for the common ratio.
Q: Can there be more than one geometric mean between two numbers?
A: Yes — you can insert as many geometric means as needed (n means), though ‘the geometric mean’ (singular) always refers to just one specific value.
Notes prepared for Punjab Board (PECTAA) 1st Year Mathematics, 2026-27 session, Chapter 6: Sequences and Series, Exercise 6.7.