Patterns of Numbers
Sequences
A sequence is an ordered list of numbers. Each number in the list is called a term.
Arithmetic Sequence: A sequence where the difference between consecutive terms is constant. This constant is called the common difference (d).
Example: 2, 5, 8, 11, 14, ... (common difference d = 3).
Formula for the nth term: aₙ = a₁ + (n-1)d
aₙ is the nth term.
a₁ is the first term.
Geometric Sequence: A sequence where the ratio between consecutive terms is constant. This constant is called the common ratio (r).
Example: 3, 6, 12, 24, 48, ... (common ratio r = 2).
Formula for the nth term: aₙ = a₁ * rⁿ⁻¹
Series
A series is the sum of the terms of a sequence.
Sum of a Finite Arithmetic Series: Sₙ = (n/2)(a₁ + aₙ)
The sum of n terms is n divided by 2, times the sum of the first and last terms.
Sum of a Finite Geometric Series: Sₙ = a₁ * (1 - rⁿ) / (1 - r)
This formula is very useful for problems involving repeated multiplication, like compound interest.
Sum of an Infinite Geometric Series: If the absolute value of the common ratio r is less than 1 (i.e., -1 < r < 1), the series converges to a finite sum.
S = a₁ / (1 - r)
If |r| ≥ 1, the series diverges and does not have a finite sum.