Sequences and Series — Precalculus
Arithmetic and geometric sequences, series formulas, sigma notation, and the convergence of infinite geometric series. Everything you need to master Chapter 12.
Quick Reference — Key Formulas
Arithmetic
- aₙ = a₁ + (n − 1)d
- Sₙ = n/2 · (a₁ + aₙ)
- Sₙ = n/2 · (2a₁ + (n−1)d)
Geometric
- aₙ = a₁ · rⁿ⁻¹
- Sₙ = a₁(1 − rⁿ) / (1 − r)
- S∞ = a₁ / (1 − r), |r| < 1
Arithmetic Sequences
An arithmetic sequence has a constant difference between consecutive terms. Add (or subtract) the same number each time.
Example: 3, 7, 11, 15, 19, ...
Common difference d = 4 (add 4 each time)
a₁ = 3, a₂ = 7, a₅ = 3 + 4(4) = 19 ✓
| Formula | Expression | Notes |
|---|---|---|
| General term | aₙ = a₁ + (n − 1)d | d = common difference |
| Common difference | d = aₙ − aₙ₋₁ | constant between consecutive terms |
| Sum of n terms | Sₙ = n/2 × (a₁ + aₙ) | need first and last terms |
| Sum (alternate) | Sₙ = n/2 × (2a₁ + (n−1)d) | need first term and d |
Worked Example — Arithmetic
Find the 20th term and the sum of the first 20 terms: 5, 9, 13, 17, ...
Step 1: Identify a₁ = 5, d = 4
Step 2: a₂₀ = 5 + (20 − 1)(4) = 5 + 76 = 81
Step 3: S₂₀ = 20/2 × (5 + 81) = 10 × 86 = 860
Check: S₂₀ = 20/2 × (2(5) + 19(4)) = 10 × 86 = 860 ✓
Geometric Sequences
A geometric sequence has a constant ratio between consecutive terms. Multiply by the same number each time.
Example: 2, 6, 18, 54, 162, ...
Common ratio r = 3 (multiply by 3 each time)
a₁ = 2, a₅ = 2 · 3⁴ = 2 · 81 = 162 ✓
| Formula | Expression | Notes |
|---|---|---|
| General term | aₙ = a₁ · rⁿ⁻¹ | r = common ratio |
| Common ratio | r = aₙ / aₙ₋₁ | constant ratio of consecutive terms |
| Sum of n terms | Sₙ = a₁(1 − rⁿ) / (1 − r) | r ≠ 1 |
| Infinite sum | S = a₁ / (1 − r) | only if |r| < 1 |
Worked Example — Geometric
Find S₆ for the sequence: 3, 6, 12, 24, ...
Step 1: a₁ = 3, r = 2
Step 2: S₆ = 3(1 − 2⁶) / (1 − 2)
Step 3: = 3(1 − 64) / (−1) = 3(−63) / (−1) = 189
Verify: 3 + 6 + 12 + 24 + 48 + 96 = 189 ✓
Infinite Geometric Series
When |r| < 1, a geometric series converges to a finite sum as you add infinitely many terms. When |r| ≥ 1, the series diverges.
Example: 1 + ½ + ¼ + ⅛ + ... (r = ½)
S = 1 / (1 − ½) = 2
Example: 1 + 2 + 4 + 8 + ... (r = 2)
Terms grow → sum keeps increasing → diverges
Worked Example — Infinite Series
Find the sum: 0.3 + 0.03 + 0.003 + 0.0003 + ...
Step 1: a₁ = 0.3, r = 0.1 (|r| = 0.1 < 1, so converges)
Step 2: S = 0.3 / (1 − 0.1) = 0.3 / 0.9 = 1/3
Note: 0.333... = 1/3 — this shows how repeating decimals are infinite geometric series ✓
Sigma Notation
Sigma notation (Σ) is shorthand for writing sums. The index tells you where to start and stop; the expression tells you what to add.
Reading sigma notation:
i = index (starts at bottom number, ends at top number)
| Rule | Formula |
|---|---|
| Constant factor | Σcaₙ = c·Σaₙ |
| Sum rule | Σ(aₙ + bₙ) = Σaₙ + Σbₙ |
| Constant sum | Σc = c·n |
| Sum of integers 1 to n | Σi = n(n+1)/2 |
| Sum of squares 1 to n | Σi² = n(n+1)(2n+1)/6 |
Worked Example — Sigma
Evaluate: Σᵢ₌₁⁵ (3i + 1)
Method 1 (expand): (4) + (7) + (10) + (13) + (16) = 50
Method 2 (split): Σ3i + Σ1 = 3·Σi + 5(1)
= 3 · (5·6/2) + 5 = 3(15) + 5 = 45 + 5 = 50 ✓
Recursive vs. Explicit Formulas
Recursive Formula
Defines each term using the previous term. You must know one term to find the next.
Arithmetic with d = 4, starting at 3
Explicit Formula
Finds any term directly from n — no need to know previous terms.
Explicit is almost always preferred for large n
Frequently Asked Questions
What is the difference between a sequence and a series?
A sequence is a list of numbers in a specific order: 2, 5, 8, 11, ... A series is the sum of the terms in a sequence: 2 + 5 + 8 + 11 + ... Sequences describe patterns; series calculate totals. On the precalculus exam, you'll need to find specific terms (sequence problems) and find sums (series problems).
How do you find the sum of an arithmetic series?
The sum of the first n terms of an arithmetic series is Sₙ = n/2 × (a₁ + aₙ), where a₁ is the first term and aₙ is the last term. You can also write it as Sₙ = n/2 × (2a₁ + (n-1)d) where d is the common difference. Example: sum of first 10 terms with a₁=3 and d=4: S₁₀ = 10/2 × (2(3) + 9(4)) = 5 × 42 = 210.
When does an infinite geometric series converge?
An infinite geometric series converges (has a finite sum) only when |r| < 1, where r is the common ratio. When it converges, S = a₁ / (1 - r). If |r| ≥ 1, the series diverges (the sum is infinite). For example, the series 1 + 1/2 + 1/4 + 1/8 + ... has r = 1/2, so it converges to S = 1/(1 - 1/2) = 2.
Practice sequences and series problems
Work through arithmetic sequences, geometric series, sigma notation, and infinite series with step-by-step solutions. Free to start.
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