A sequence is an ordered list of numbers: a₁, a₂, a₃, … Each position n has a term aₙ. A series is what you get when you add terms—often a partial sum of the first N terms.
Arithmetic sequences: steady steps
Each term adds the same fixed amount—the common difference d. Example: 3, 7, 11, 15, … (d = 4). The nth term is:
aₙ = a₁ + (n − 1)·d
Think of climbing stairs of equal height, or a salary that rises by the same dollar amount each year.
Geometric sequences: steady multipliers
Each term multiplies by the same common ratio r. Example: 2, 6, 18, 54, … (r = 3). The nth term is:
aₙ = a₁ · rⁿ⁻¹
This models doubling investments, bacteria counts, or a photo resized by the same scale factor repeatedly.
Partial sums
Adding the first N arithmetic terms has a neat formula involving first and last term. Geometric partial sums use a formula with r (when r ≠ 1). In the lab you practice reading the pattern and computing specific terms and small sums by hand so the formulas have meaning.
Common trap: Mixing “add d” with “multiply by r.” If the gaps between terms grow, it is usually geometric, not arithmetic.