Exercise 6.7 Class 11 Math Solutions - Unit 6 Sequences and Series, Punjab Board
Step-by-step solutions to Exercise 6.7, Unit 6: Sequences and Series, for Class 11 Punjab Board students. Covers the sum of n terms of a finite Geometric Series Sₙ = a₁(1 - rⁿ)/(1 - r), evaluating infinite Geometric Series S_∞ = a₁/(1 - r) for |r| < 1, and converting recurring decimals to rational fractions worked out question by question.
Exercise Overview
Exercise 6.7 in Unit 6 of Class 11 Mathematics explores Geometric Series-the indicated summation of terms in a Geometric Progression: Sₙ = a₁ + a₁r + a₁r² + ... + a₁rⁿ⁻¹. Students apply the finite summation formula Sₙ = a₁(1 - rⁿ)/(1 - r) to compute exact partial sums, determine missing terms or common ratios from known series values, and analyze growth behavior.
A central focus of the exercise is the theory of infinite Geometric Series. When the common ratio satisfies |r| < 1, the term rⁿ approaches zero as n tends to infinity, allowing the infinite sum to converge to a finite value S_∞ = a₁ / (1 - r). Students use this fundamental convergence theorem to solve classic mathematical problems, such as expressing recurring decimal fractions (e.g., 0.3̇3̇ or 0.2̇3̇) as precise common fractions in simplest form p/q, as well as modeling physical rebound distances of bouncing objects.
In Punjab Board HSSC Part 1 examinations, Exercise 6.7 is a consistent favorite for Section B short questions (infinite geometric series and recurring decimals) and Section C long questions (sum of n terms algebraic proofs) across Lahore, Rawalpindi, Gujranwala, Faisalabad, and Multan boards. Practicing these step-by-step solutions ensures students develop full mastery over geometric series summation techniques.