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Learn · Pre-Calculus

Sequences and Series: Arithmetic, Geometric, and Sigma

The four formulas that handle every Pre-Calc sequence question — plus the convergence rule that decides whether an infinite sum exists.

7 minTEKS 5A-5CPre-Calculus

Arithmetic sequences

Each term differs from the previous by a constant d (common difference). nth term: aₙ = a₁ + (n − 1)d. Sum of first n: Sₙ = n(a₁ + aₙ)/2 = n/2 · [2a₁ + (n − 1)d].

Arithmetic: constant difference d. Geometric: constant ratio r.Arithmetic (d = 3)2581114+3+3+3+3aₙ = a₁ + (n−1)·dGeometric (r = 2)36122448×2×2×2×2aₙ = a₁·r^(n−1)

Geometric sequences

Each term is a constant r times the previous. nth term: aₙ = a₁ · r^(n−1). Finite sum: Sₙ = a₁ (1 − rⁿ)/(1 − r).

Series = sum of terms — closed-form formulas for arithmetic & geometricArithmetic partial sumSₙ = n/2 · (a₁ + aₙ)n terms × average of first + laste.g., 1+2+…+100 = 100/2·(1+100) = 5050Geometric partial sumSₙ = a₁ · (1 − rⁿ)/(1 − r)infinite if |r|<1: S∞ = a₁/(1 − r)e.g., ½ + ¼ + ⅛ + … = 1

Infinite geometric series

Sum converges iff |r| < 1. Then: S∞ = a₁ / (1 − r).

📐 Example

1 + 1/2 + 1/4 + 1/8 + ... has r = 1/2, |r| < 1 → S = 1/(1 − 1/2) = 2. But 1 + 2 + 4 + 8 + ... has r = 2 ≥ 1 → diverges (sum is ∞, no value).

Sigma notation: compact way to write a sumΣnk = 1aₖa₁ + a₂ + a₃ + … + aₙexample: Σ (k=1 to 4) k² = 1 + 4 + 9 + 16 = 30

Sigma notation

Σ_{k=1}^{n} f(k) means "sum f(k) for k = 1, 2, ..., n". Useful identities:

  • Σ k = n(n+1)/2
  • Σ k² = n(n+1)(2n+1)/6
  • Σ c = c · n (constant)
  • Σ [a·f(k) + b·g(k)] = a·Σf + b·Σg (linearity)

Check yourself

📌 Compute
Sum the infinite series 3 + 1 + 1/3 + 1/9 + ...

Practice with CBE-style practice questions

Pre-Calc Sem B practice for sequences and series.