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.
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].
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).
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
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 + ...
a₁ = 3, r = 1/3. S = 3/(1 − 1/3) = 3/(2/3) = 9/2.
Practice with real CBE questions
Pre-Calc Sem B practice for sequences and series.