Formula and Method for Summing a Linear Number Sequence
A linear number sequence — better known as an arithmetic sequence — is a list of numbers in which each term is found by adding the same fixed amount, the common difference (d), to the previous term. The nth term is given by aₙ = a₁ + (n − 1)d, and the sum of the first n terms is Sₙ = n/2 × (2a₁ + (n − 1)d), which is the same as multiplying the number of terms by the average of the first and last term.
How the calculation works
Enter the first term (a₁), the common difference (d), and the number of terms (n). The calculator first finds the last term with aₙ = a₁ + (n − 1)d, then plugs that into Sₙ = n × (a₁ + aₙ) / 2 to get the total sum. This shortcut — pairing the first and last term, the second and second-to-last, and so on — is the same trick attributed to a young Carl Friedrich Gauss, who reportedly summed 1 through 100 by noticing that 50 pairs each add up to 101.
Common mistakes
- Off-by-one errors in n: the exponent in aₙ = a₁ + (n − 1)d uses (n − 1), not n — the first term already counts as term 1, so you add the difference (n − 1) times, not n times.
- Forgetting a negative common difference: if d is negative the sequence decreases (e.g., 20, 15, 10, ...), and the sum can come out negative once enough terms are added — that is expected, not an error.
- Confusing the sum with the last term: Sₙ is the total of every term added together, while aₙ is just the value of the final term — they are related but not the same number.
Real-world applications
- Counting seats in a theater or stadium section where each row has a fixed number more seats than the last.
- Stacking problems, such as cans or boxes arranged in a triangular pile with one fewer item per row.
- Simple savings or repayment plans where a fixed amount is added (or subtracted) each period.
- Any pattern of evenly-spaced numbers, such as scheduling, page counts, or step-based measurements.