Pre-Calculus: Sequences and Series Formulas

Arithmetic Explicit Formula

aₙ=a₁+d(n-1)

Arithmetic Recursive Formula

a₁=___; aₙ=aₙ₋₁+d

Arithmetic Series Formula

Sₙ=n/2(a₁+aₙ)

Geometric Explicit Formula

aₙ=a₁(rⁿ⁻¹)

Geometric Recursive Formula

a₁=___; aₙ=aₙ₋₁(r)

Geometric Finite Formula

Sₙ=a₁(1-rⁿ/1-r)

Geometric Infinite Formula

S∞=a₁(1/1-r)

Definition of Sequence

an ordered list of numbers

Common notation for sequences

subscript notation

Arithmetic or Geometric?

2, 4, 5, 6, 10,...

Arithmetic

Definition of Arithmetic Sequence

ordered list of numbers in which the difference (d) of consecutive terms is constant

Arithmetic or Geometric?

linear relationship

Arithmetic

Definition of Explicit Formula

used to calculate any term in the sequence as long as the first term is known

Arithmetic or Geometric?

constant rate of change

Arithmetic

Definition of Recursive Formula

a rule where all terms of the sequence are defined using the previous term

Definition of Series

the sum of the terms in a sequence

Definition of Partial Sum:

the sum of a specified number of terms in a sequence

Common notation for series

sigma notation

Definition of Geometric Sequence

sequence in which the ratio between any two consecutive terms is constant

Arithmetic or Geometric?

common ratio

geometric

Arithmetic or Geometric?

3, 12, 48, 192,...

geometric

Arithmetic or Geometric?

exponential relationship

Geometric

Definition of Geometric Series

the sum of a geometric sequence

If _____________, the geometric series is convergent

|r| < 1

If ____________, the geometric series is divergent

|r| > 1