How this instrument works
A linear number sequence is called that because its terms sit on a straight line when you plot value against position: term k is a₁ + (k−1)d, so k and the term move together at the constant rate d, exactly like y = mx + b. This sheet does not need to list a₁, a₂, a₃… out to the nth entry to find their total; the closed-form S = n⁄2 × (2a₁ + (n−1)d) reaches the total in one multiplication and one addition, however large n gets.
The 2 in the denominator is an averaging step in disguise. Sum equals count times average, S = n × (mean term), and because the terms are evenly spaced, the mean of the whole run equals the mean of just its two ends, (a₁+aₙ)⁄2 — a fact usually credited to a schoolboy Gauss, said to have noticed that the first and last, the second and second-last, and every such pair share that identical average. Swap in aₙ = a₁+(n−1)d for the last term and the formula this calculator uses drops straight out.
Hold a₁ and d fixed and watch what happens as n grows: because the last term aₙ itself grows with n, the sum does not scale in step with the number of terms — it scales roughly with n², so doubling how many terms you add typically more than doubles the total, unless d is zero. Set d to zero and every term equals a₁, so the sum collapses to the plain product n×a₁, the one case where the sequence stops being genuinely linear-in-position and the squared term vanishes entirely.
- Enter the sequence's starting value into First term, a₁.
- Enter the fixed step between consecutive terms into Common difference, d — use a negative number for a sequence that counts down.
- Enter how many terms to add into Number of terms, n; it should be a positive whole number.
- Read Sum of the sequence for the total — every term from the first through the nth is already added in, with none listed individually.
Worked example — five rows starting at 2, stepping by 3
Picture a market stall stacking crates in five rows behind the counter: the front row holds 2 crates, and each row further back holds 3 more than the row in front of it, giving 2, 5, 8, 11, and 14 crates across the five rows. Set First term, a₁ to 2, Common difference, d to 3, and Number of terms, n to 5, and Sum of the sequence returns S = 5⁄2 × (2×2 + (5−1)×3) = 2.5 × (4+12) = 2.5 × 16 = 40 crates in total, without counting a single row by hand.
Direct addition confirms it: 2+5+8+11+14 = 40. The shortcut works because the two outer rows form a matched pair worth 16 apiece — the front row's 2 with the back row's 14, and the second row's 5 with the second-to-last row's 11 — while the middle row of 8 sits exactly halfway between them; two such pairs plus that lone middle term give 16+16+8 = 40, the same total the formula reached directly from a₁, d, and n alone.
Questions
What does 'linear' mean in a linear number sequence?
It means each term sits on a straight line when plotted against its position: term k equals a₁+(k−1)d, the same shape as y = mx + b with slope d. Because the step between neighbors never changes, the sequence is also called arithmetic — 'linear' just names the same fixed-step pattern by its graph instead of its algebra.
How is the sum formula actually derived?
Sum equals count times average: S = n × (mean of the n terms). Because the terms are evenly spaced, that mean equals the average of just the first and last term, (a₁+aₙ)⁄2, so S = n⁄2 × (a₁+aₙ). Substituting aₙ = a₁+(n−1)d for the last term turns that into S = n⁄2 × (2a₁+(n−1)d), the exact formula this calculator evaluates.
Why does the total grow quadratically instead of in step with the number of terms?
Because the last term aₙ = a₁+(n−1)d itself grows as n grows, and the sum is n times an average that includes aₙ; multiplying two quantities that both grow with n produces an n² term. Expand the formula and S = (d⁄2)n² + (a₁−d⁄2)n — doubling n roughly quadruples the total once that squared term dominates, unless d is zero.
Can the common difference or first term be negative or a fraction?
Yes — the formula places no restriction on either. A negative common difference produces a descending run, such as a balance that shrinks by a fixed amount every period, and both a₁ and d can carry decimals; the arithmetic works identically, only the sign and size of the numbers change.
How is summing this sequence different from summing a geometric one?
This calculator totals a run built by repeated addition of a fixed difference d, and the total grows quadratically with the number of terms. A geometric sequence is instead built by repeated multiplication by a fixed ratio; its sum, covered by the separate sum-of-series calculator on this site, can grow exponentially or, for a ratio between −1 and 1, settle toward a finite limit — a completely different shape of growth from a straight fixed step.