Chapter 5: Progressions

5.1 Arithmetic Progressions

• An arithmetic progression (AP) is a sequence in which each term is obtained by adding a constant to the previous term. (p.129)
• Common difference d = T₂ − T₁ = T₃ − T₂ = ... = T_n − T_n−1. (p.128)
• nth term: T_n = a + (n − 1)d. (p.131)
• Sum of first n terms: S_n = (n/2)[2a + (n − 1)d]. (p.133)
• S_n = (n/2)(a + l), where l is the last term. (p.134)
• T_n = S_n − S_n−1. (p.134)
• Arithmetic progression is written T₁, T₂, ...; arithmetic series is T₁ + T₂ + .... (p.135)

5.2 Geometric Progressions

• A geometric progression (GP) is a sequence in which each term is obtained by multiplying the previous term by a constant. (p.140)
• Common ratio r = T₂/T₁ = T₃/T₂ = ... = T_n/T_n−1. (p.140)
• nth term: T_n = arⁿ⁻¹. (p.142)
• Sum of first n terms: S_n = a(1 − rⁿ)/(1 − r) (commonly used for |r| < 1) or S_n = a(rⁿ − 1)/(r − 1) (commonly used for |r| > 1), r ≠ 1. (p.144)
• T_n = S_n − S_n−1 also holds for geometric progressions. (p.144)
• Sum to infinity: S_∞ = a/(1 − r), where |r| < 1 (since rⁿ → 0 as n → ∞). (p.146)
• Recurring decimals can be written as a geometric series and converted to fractions using S_∞. (p.147)