Chapter 7. Sequences and Series

Arithmetic sequence: a constant difference d. The nth term is aₙ = a₁ + (n − 1)d. The sum of the first n terms is Sₙ = n(a₁ + aₙ)/2 [3][11].

Geometric sequence: a constant ratio r. The nth term is aₙ = a₁rⁿ⁻¹. The sum of the first n terms is Sₙ = a₁(1 − rⁿ)/(1 − r) for r ≠ 1. If |r| < 1, the infinite sum is a₁/(1 − r).

Sigma notation. Σ (from i = 1 to n) of aᵢ means a₁ + a₂ + … + aₙ.

Binomial theorem. (a + b)ⁿ = Σ C(n, k) aⁿ⁻ᵏ bᵏ, where C(n, k) = n! / (k!(n − k)!).

Review: (a) Find 1 + 2 + … + 100. (Answer: 5050.) (b) Find 1 + 1/2 + 1/4 + … (Answer: 2.)