Exercise 6.5 Class 11 Math Solutions - Unit 6 Sequences and Series, Punjab Board

Step-by-step solutions to Exercise 6.5, Unit 6: Sequences and Series, for Class 11 Punjab Board students. Covers Geometric Progression fundamentals, evaluating the common ratio r, finding the nth term using aₙ = a₁ rⁿ⁻¹, solving simultaneous exponential equations, and verifying algebraic properties of GP worked out question by question.

This page provides complete, step-by-step solved solutions for Exercise 6.5 of Unit 6: Sequences and Series for Class 11 Mathematics under the Punjab Board (PCTB / PECTAA). All questions involving Geometric Progressions (GP), determining common ratio r = aₖ/aₖ₋₁, applying the general nth-term formula aₙ = a₁ rⁿ⁻¹, finding term positions, proving algebraic GP identities, and symmetric selection of terms (a/r, a, ar) are thoroughly solved for HSSC Part 1 board exam preparation.
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Exercise Overview

Exercise 6.5 introduces the comprehensive study of Geometric Progressions (GP) in Unit 6 of Class 11 Mathematics. A geometric sequence is characterized by a constant non-zero quotient between any term and its predecessor, known as the common ratio r = aₖ / aₖ₋₁. Through the fundamental exponential relation aₙ = a₁ rⁿ⁻¹, students learn to evaluate any term of the progression, determine unknown terms from given boundary conditions, and establish the total count of elements in a finite geometric sequence.

The exercise explores essential algebraic techniques, including constructing and solving simultaneous exponential equations to determine the initial term a₁ and common ratio r. Furthermore, students master symmetric parameterization of consecutive terms (such as a/r, a, ar for three terms) when products of terms are provided, enabling the elimination of r upon multiplication and simplifying complex polynomial systems into basic radicals.

In Punjab Board HSSC Part 1 examinations, Exercise 6.5 is heavily featured in Section B short questions (finding specified GP terms) and Section C long questions (solving product-sum word problems and proving theoretical GP relationships) across Lahore, Rawalpindi, Gujranwala, Faisalabad, and Multan boards. These step-by-step solutions equip students with clear mathematical rigor and high examination precision.

FAQs

Are these Exercise 6.5 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 all Geometric Progression concepts, nth-term rules, and properties for the 2026-2027 academic session.
Do these solutions cover every question in Exercise 6.5?
Yes, every textbook problem from Exercise 6.5 is completely solved with step-by-step algebraic working, including finding missing terms in a GP, determining the common ratio r, and proving GP relationships.
What is the formula for the general nth term of a Geometric Progression?
The general nth term of a Geometric Progression is given by aₙ = a₁ * rⁿ⁻¹, where a₁ is the first term, r is the common ratio (r = aₖ / aₖ₋₁ ≠ 0), and n is the term position (positive integer).
How do you select consecutive terms in a GP when their product is given?
To simplify calculations: for 3 terms, choose a/r, a, ar; for 4 terms, choose a/r³, a/r, ar, ar³; for 5 terms, choose a/r², a/r, a, ar, ar².
Can students read these solutions online?
Yes, students can read all Exercise 6.5 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 specific terms of a GP, determining common ratios, and solving GP algebraic proofs are frequently featured in Section B short questions and Section C long questions for HSSC Part 1.
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 6.5 solutions on MaryamNotes.pk are available entirely free of cost for students and teachers across Punjab.