Rucete ✏ AP Precalculus In a Nutshell
6. Sequences
This chapter introduces sequences as ordered lists of numbers based on patterns or rules. You’ll explore arithmetic and geometric sequences, learn how to find general terms, and understand how these relate to linear and exponential functions.
- Introduction to Sequences
- Sequence: a list of numbers in a specific order
- Can be finite or infinite
- Can be expressed as functions: input = n, output = aₙ
- Examples:
- Arithmetic: common difference (e.g., 1, 3, 5, 7, ...)
- Geometric: common ratio (e.g., 2, 4, 8, 16, ...)
- Recursive: uses previous terms (e.g., Fibonacci)
- Domain and Range of Sequences
- Domain = whole numbers (n = 1, 2, 3, ...)
- Range = real numbers (term values)
- Arithmetic Sequences
- Terms increase/decrease by a constant difference d
- General Formulas:
- If first term is a₁: aₙ = a₁ + (n − 1)d
- If starting from a₀: aₙ = a₀ + nd
- If position unknown: aₙ = aₖ + (n − k)d
- Example: a₁ = 0.5, d = −1.5 → aₙ = 0.5 − 1.5(n − 1)
- Average rate of change is constant
- Geometric Sequences
- Each term is multiplied by a constant ratio r
- General Formulas:
- If first term is g₁: gₙ = g₁ · rⁿ⁻¹
- If starting from g₀: gₙ = g₀ · rⁿ
- If gₖ is known: gₙ = gₖ · rⁿ⁻ᵏ
- Example: g₁ = 36, g₅ = 324 → solve for r, find g₆
- Non-constant rate of change—growth by factor
- Comparing Sequences to Functions
- Arithmetic ↔ Linear functions
- aₙ = a₁ + (n − 1)d ↔ y = mx + b
- Constant addition
- Geometric ↔ Exponential functions
- gₙ = g₁ · rⁿ⁻¹ ↔ f(x) = abˣ
- Constant multiplication
- Differences:
- Sequences → domain is whole numbers (discrete)
- Functions → domain is real numbers (continuous)
- Similar formats:
- Linear: y = y₁ + m(x − x₁)
- Arithmetic: aₙ = aₖ + (n − k)d
- Exponential: f(x) = abˣ or a · r^(x−x₁)
- Example: Identifying Types
f(n) = −3 + 4n → Arithmetic (linear form)
h(n) = 4(−3.5)ⁿ⁻¹ → Geometric (exponential form)
- Identify by growth pattern:
- Equal spacing → arithmetic
- Rapid change → geometric
- Domain and Range Differences
- Sequence:
- Domain = specific n-values (e.g., {1, 2, 3, 4, 5})
- Range = corresponding outputs
- Generating function:
- Domain = all real numbers
- Range depends on type (linear or exponential)
In a Nutshell
Sequences are ordered lists of numbers with predictable patterns. Arithmetic sequences grow by addition, while geometric ones grow by multiplication. Their equations parallel linear and exponential functions but are restricted to discrete values. Recognizing, writing, and comparing these sequences sharpens both algebraic and functional thinking in precalculus.