Geometric Progression

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A Geometric Progression is a sequence where the ratio of any term to its preceding term is constant. This constant ratio is called the common ratio (r). Example: 3, 6, 12, 24, ... Here, 6/3 = 2, 12/6

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The nth term of a G.P. is given by: Tₙ = arⁿ⁻¹ Where: - a = first term - r = common ratio - n = position of the term Example: For G.P. 2, 6, 18, 54, ... T₄ = 2 × 3³ = 2 × 27 = 54

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Given: a = 4, r = 12/4 = 3 Using Tₙ = arⁿ⁻¹ T₅ = 4 × 3⁴ T₅ = 4 × 81 T₅ = 324 Therefore, the 5th term is 324.

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A Geometric Series is the sum of terms of a Geometric Progression. If G.P. is: a, ar, ar², ar³, ... Then G.S. is: a + ar + ar² + ar³ + ... Difference: - G.P. is a sequence (ordered list) - G.S. is t

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When r ≠ 1: Sₙ = a(rⁿ - 1)/(r - 1) Alternative form: Sₙ = (rTₙ - a)/(r - 1) Where: - a = first term - r = common ratio - n = number of terms - Tₙ = nth term

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When r = 1: Sₙ = na Where: - n = number of terms - a = first term (and all terms are equal) Example: G.P. 5, 5, 5, 5, ... S₄ = 4 × 5 = 20

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Given: a = 2, r = 6/2 = 3, n = 6 Using Sₙ = a(rⁿ - 1)/(r - 1) S₆ = 2(3⁶ - 1)/(3 - 1) S₆ = 2(729 - 1)/2 S₆ = 2 × 728/2 S₆ = 728 Therefore, sum of first 6 terms is 728.

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Three consecutive terms of a G.P. can be represented as: a/r, a, ar This form simplifies calculations when: - Sum and product of three terms are given - Middle term is known Example: If three terms