Worked examples

07 Worked examples

Calculating Sum of Terms and Sum to Infinity of a G.P.Problem Statement & Given Data: In a geometric sequence, the first term is 32 and the third term is 8. If all terms are positive, find: (1) The common ratio r, (2) The 6th term T6, (3) The sum of the first 6 terms S6, (4) The sum of the series to infinity S_inf.Step-by-Step Mathematical & Chemical Solution: [1] Given: First term a = 32. Positive terms ==> r > 0. [2] Third term T3 = a r² = 8. [3] 32 r² = 8 ==> r² = 8 / 32 = 1/4. [4] Since terms are positive: r = +√(1/4) = 1/2. [5] Step 1: The common ratio r is 1/2. [6] Step 2: Calculate the 6th term T6: T6 = a r⁵ = 32 × (1/2)⁵ = 32 × (1/32) = 1. [7] Step 3: Calculate the sum of the first 6 terms S6: Sn = a(1 - rⁿ) / (1 - r). [8] S6 = [32 × (1 - (1/2)⁶)] / (1 - 1/2) = [32 × (1 - 1/64)] / (1/2) = 64 × (63 / 64) = 63. [9] Step 4: Calculate the sum to infinity S_inf (condition |r| = 1/2 < 1 is satisfied): [10] S_inf = a / (1 - r) = 32 / (1 - 1/2) = 32 / (1/2) = 64.💡 High-Yield Exam Shortcut & Pitfall Alert: Exam Tip: The sum to infinity is only valid when |r| < 1. If r = 2, S_inf does not exist!