Exercise 6.9 Class 11 Math Solutions - Unit 6 Sequences and Series, Punjab Board

Step-by-step solutions to Exercise 6.9, Unit 6: Sequences and Series, for Class 11 Punjab Board students. Covers Harmonic Progression fundamentals, finding the nth term via reciprocal arithmetic progressions, calculating the single Harmonic Mean H = 2ab/(a + b), and inserting n Harmonic Means between given numbers worked out question by question.

This page provides complete, step-by-step solved solutions for Exercise 6.9 of Unit 6: Sequences and Series for Class 11 Mathematics under the Punjab Board (PCTB / PECTAA). All questions on the definition and general nth term of a Harmonic Progression (HP), computing the single Harmonic Mean H = 2ab/(a + b), and inserting n Harmonic Means H₁, H₂, ..., Hₙ between given endpoints via reciprocal AP transformation are thoroughly worked out for HSSC Part 1 board exam preparation.
Back
Exercise 6.8
Next
Exercise 6.10
Advertisement

Exercise Overview

Exercise 6.9 in Unit 6 of Class 11 Mathematics investigates Harmonic Progressions (HP) and Harmonic Means (HM). A sequence of non-zero numbers is defined as a Harmonic Progression if the sequence of their reciprocals forms an Arithmetic Progression (AP). Because no direct summation formula exists for general harmonic series, all analytical operations-such as identifying terms, calculating common differences, or finding term positions-are performed by inverting the sequence into its AP counterpart.

The exercise thoroughly covers the single Harmonic Mean between two non-zero real numbers a and b, establishing the formula H = 2ab / (a + b). It then extends this concept to inserting n intermediate harmonic means H₁, H₂, ..., Hₙ between a and b. Students learn to convert the endpoints into reciprocals 1/a and 1/b, insert n arithmetic means with common difference d = (1/b - 1/a)/(n + 1), and invert each intermediate arithmetic mean to obtain the exact harmonic means Hₖ = 1 / Aₖ.

In Punjab Board HSSC Part 1 examinations, Exercise 6.9 is frequently featured in Section B short questions (finding specified HP terms or inserting harmonic means) and Section C long questions (solving reciprocal progression proofs) across Lahore, Rawalpindi, Faisalabad, Gujranwala, and Multan boards. Mastering these step-by-step solutions gives students confidence and mathematical precision in dealing with fractional progressions.

FAQs

Are these Exercise 6.9 solutions according to the Punjab Board syllabus?
Yes, these solutions strictly follow the official Punjab Curriculum and Textbook Board (PCTB / PECTAA) syllabus for Class 11 Mathematics, covering Harmonic Progressions and Harmonic Means for the 2026-2027 academic session.
Do these solutions cover every question in Exercise 6.9?
Yes, every textbook problem from Exercise 6.9 is completely solved with step-by-step algebraic working, including finding missing terms of HP, evaluating single harmonic means, and inserting multiple harmonic means.
What is a Harmonic Progression (HP)?
A sequence of non-zero numbers is called a Harmonic Progression if the reciprocals of its terms form an Arithmetic Progression (AP). Its general nth term is given by aₙ = 1 / [1/a₁ + (n - 1)d].
What is the formula for the Harmonic Mean between two numbers a and b?
The single Harmonic Mean H between two non-zero numbers a and b is given by H = (2ab) / (a + b).
How do you insert n Harmonic Means between two numbers a and b?
To insert n Harmonic Means H₁, H₂, ..., Hₙ between a and b, first insert n Arithmetic Means A₁, A₂, ..., Aₙ between 1/a and 1/b with common difference d = (1/b - 1/a)/(n + 1). Then, the k-th harmonic mean is Hₖ = 1 / Aₖ.
Can students read these solutions online?
Yes, students can read all Exercise 6.9 solutions directly online through MaryamNotes.pk on any web browser without needing to download external files or install third-party apps.
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 6.9 solutions on MaryamNotes.pk are available entirely free of cost for students and educators across Punjab.