Theory of Geometric Progressions (G.P.) and Convergence to Infinity

A Geometric Progression (G.P.) is characterized by an invariant multiplicative factor: T_(n+1) / T_n = r (constant common ratio, r ≠ 0). General Term: T1 = a, T2 = ar, T3 = ar², giving: T_n = a · r^{n - 1}.

Partial Sum Formula: Multiplying the partial sum S_n = a + ar + ar² + ... + ar^n-1 by r and subtracting gives: S_n - r S_n = a - a rⁿ -> S_n = [a(1 - rⁿ)] / (1 - r) = (a - l r) / (1 - r), where r ≠ 1.

Infinite Series & Convergence: What happens when the number of terms n approaches infinity? If |r| < 1 (i.e., -1 < r < 1), the term rⁿ decays asymptotically to zero: lim_{n -> ∞} rⁿ = 0. Therefore, the infinite sum converges to: S_∞ = lim_{n -> ∞} S_n = a / (1 - r). If |r| ≥ 1, rⁿ diverges, and the sum to infinity does NOT exist.

Geometric Means: Inserting k geometric means between A and B produces a G.P. of n = k + 2 terms. Common ratio: r = (B / A)^{1 / (k + 1)}.