Sequences and Series

A sequence where the ratio

A sequence where the ratio of any term to its preceding term is constant. For a G.P. with first term 'a' and common ratio 'r', the terms are: a, ar, a

For finite G

For finite G.P.: Sn = a(r^n - 1)/(r - 1) when r ≠ 1, and Sn = na when r = 1. For infinite G.P.: S∞ = a/(1-r) when |r| < 1. Example: For G.P. 3, 6, 12,

A sequence where the reciprocals

A sequence where the reciprocals of terms form an A.P. If 1/t₁, 1/t₂, 1/t₃... are in A.P., then t₁, t₂, t₃... are in H.P. Example: 1/2, 1/5, 1/8, 1/11

For two positive numbers x

For two positive numbers x and y: A.M. = (x+y)/2, G.M. = √(xy), H.M. = 2xy/(x+y). Important relationship: H ≤ G ≤ A, with equality when x = y. Also, G

Uses the symbol Σ to represent

Uses the symbol Σ to represent sums. Key formulas: Σr = n(n+1)/2, Σr² = n(n+1)(2n+1)/6, Σr³ = [n(n+1)/2]². Properties include: Σ(ar + b) = aΣr + nb, w