Quick answer
Sequences list terms; series add them. Formulas target different questions even when the same numbers appear.
Formula
- Sequence: aₙ = a₁ + (n - 1)d
- Series: Sₙ = n/2 (a₁ + aₙ)
Introduction
Mixing sequence and series language is one of the fastest ways to lose points on otherwise correct work. The symbols look similar, but the questions diverge.
Definitions live in what is an arithmetic sequence; sums are developed in the arithmetic series formula article.
Train yourself to underline verbs: "find the 8th term" versus "find the total of the first 8 terms." That habit prevents formula swaps.
Definitions side by side
Arithmetic sequence: an ordered list with constant difference between consecutive terms.
Arithmetic series: the sum of terms from an arithmetic sequence, often written with plus signs or sigma notation.
You can have a sequence without immediately forming a series, but a series built from arithmetic terms relies on the sequence structure underneath.
Formula comparison
- Term: aₙ = a₁ + (n - 1)d
- Sum: Sₙ = n/2 (a₁ + aₙ)
The term formula locates one position. The sum formula compresses many additions into one expression.
If a problem gives d but not the last term, you may use the sequence formula first to find aₙ, then plug into the sum formula.
Keeping a two-column note (list vs total) on scratch paper helps during timed tests.
Step-by-step guide
- Read for keywords. Highlight sum, total, cumulative, or add versus single-term language.
- Choose the model. Sequence work lists; series work totals.
- Compute endpoints. Find a₁ and aₙ when a sum is required.
- Apply the matching formula. Do not plug list values into the sum formula without intent.
Parallel example
Sequence view: 2, 5, 8, 11, 14 is an arithmetic list with a₁ = 2 and d = 3.
Series view: 2 + 5 + 8 + 11 + 14 = 40 totals the same terms.
The list helps you identify a₁ and aₙ; the sum formula S₅ = 5/2 × (2 + 14) also yields 40.