Exercise 6.6 Class 11 Math Solutions - Unit 6 Sequences and Series, Punjab Board

Step-by-step solutions to Exercise 6.6, Unit 6: Sequences and Series, for Class 11 Punjab Board students. Covers computing the single Geometric Mean G = ±√(ab) between two numbers, inserting n Geometric Means G₁, G₂, ..., Gₙ between given endpoints, and proving that the product of n geometric means equals the nth power of the single geometric mean worked out question by question.

This page provides complete, step-by-step solved solutions for Exercise 6.6 of Unit 6: Sequences and Series for Class 11 Mathematics under the Punjab Board (PCTB / PECTAA). All questions involving the computation of the single Geometric Mean G = ±√(ab) between two numbers, inserting n Geometric Means G₁, G₂, ..., Gₙ between given endpoints, and proving fundamental algebraic identities such as ∏ Gᵢ = Gⁿ are thoroughly worked out for HSSC Part 1 board exam preparation.
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Exercise Overview

Exercise 6.6 is dedicated to the mathematical theory of Geometric Means (G.M.) within Unit 6 of Class 11 Mathematics. A number G is defined as the Geometric Mean between two non-zero real numbers a and b if the sequence a, G, b forms a Geometric Progression. This condition leads directly to the standard formula G = ±√(ab), with the sign chosen to match the signs of a and b.

The exercise broadens this foundation to the insertion of n intermediate geometric means G₁, G₂, ..., Gₙ between two endpoints a and b. This construction generates an (n + 2)-term geometric progression where the common ratio is r = (b/a)^(1/(n + 1)). Students learn to evaluate each intermediate mean via Gₖ = a · rᵏ and prove vital theoretical theorems, notably establishing that the continuous product of all n inserted geometric means is identically equal to the nth power of the single geometric mean: G₁ · G₂ ··· Gₙ = Gⁿ.

In Punjab Board HSSC Part 1 examinations, Exercise 6.6 is an essential source for Section B short questions (inserting 2, 3, or 4 geometric means) and Section C long questions (proving geometric mean product identities) across Lahore, Rawalpindi, Gujranwala, Faisalabad, and Multan boards. Practicing these step-by-step solutions gives students mastery over radicals and exponential indices for high exam scores.

FAQs

Are these Exercise 6.6 solutions according to the Punjab Board syllabus?
Yes, these solutions strictly adhere to the official Punjab Curriculum and Textbook Board (PCTB / PECTAA) syllabus for Class 11 Mathematics, covering single and n geometric means for the 2026-2027 academic session.
Do these solutions cover every question in Exercise 6.6?
Yes, every textbook problem from Exercise 6.6 is completely solved with step-by-step algebraic working, including evaluating single geometric means, inserting intermediate geometric means, and proving algebraic product theorems.
What is the formula for the Geometric Mean between two numbers a and b?
The single Geometric Mean G between two real numbers a and b of the same sign is given by G = ±√(ab).
How do you insert n Geometric Means between two numbers a and b?
If G₁, G₂, ..., Gₙ are inserted between a and b, they form a GP of (n + 2) terms with first term a and last term b. The common ratio is r = (b / a)^(1 / (n + 1)), and the k-th mean is Gₖ = a * rᵏ.
Can students read these solutions online?
Yes, students can read all Exercise 6.6 solutions directly online through MaryamNotes.pk on any web browser without needing to download external files or install third-party apps.
Are these solutions useful for Punjab Board exams?
Yes, calculating geometric means, inserting multiple means, and verifying the identity (G₁ · G₂ ··· Gₙ) = Gⁿ are frequently tested in Section B short questions and Section C long questions.
Are these solutions based on the latest PCTB / PECTAA syllabus?
Yes, they are fully aligned with the updated Student Learning Outcomes (SLO) mandated by the Punjab Curriculum and Textbook Board (PECTAA) for the 2026-2027 academic year.
Are these solutions free for students?
Yes, all Class 11 Mathematics Exercise 6.6 solutions on MaryamNotes.pk are available entirely free of cost for students and teachers across Punjab.