Sequences and Series

Arithmetic

Adding by the same number

Geometric

Multiplying by the same number

Geometric

Multiplying by the same number

Explicit arithmetic equation

an = a₁(n-1)d

Recursive arithmetic equation

aₙ = a₍ₙ₋₁₎ + d

Explicit Geometric Equation

aₙ = a₁(r⁽ⁿ⁻¹⁾)

Recursive Geometric Equation

aₙ = r(a₍ₙ₋₁₎)

Sum of Finite Series - Arithmetic

Sum = n((a₁+aₙ)/2)

Sum of Finite Series - Geometric

Sum = (a₁(1-r)ⁿ)/(1-r)

Sum of Infinite Series - Arithmetic

Sum = no sum because the graph would form a line that has no end.

Sum of Infinite Series - Geometric

Sum = If |r| < 1, series converges: a₁/(1-r)

Sum of Infinite Series - Geometric

Sum = If |r| > 1, series diverges: No sum

Constant Multiple of Sigma Notation

You can factor a constant out of a sigma

Sum/Difference of Sigma Notation

You can split a sigma across addition and subtraction

Constant term

Summing a constant n time gives n times c