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Category Math & Algebra

First Term (a₁) Any real number

Common Difference (d) Constant step between terms

Number of Terms (n) Positive integer, max 1000

Find Specific Term (k) optional Index of specific term to find

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About

An arithmetic sequence (arithmetic progression) is a sequence of numbers where the difference between consecutive terms remains constant. This constant is the d (common difference). Miscalculating d or the sum formula propagates errors through financial amortization schedules, signal sampling intervals, and structural load distributions. This tool computes the k-th term, partial sum S_n, and generates the full term table using the closed-form Gauss formula. It assumes integer or decimal inputs and does not handle symbolic expressions.

Pro tip: for sequences with thousands of terms, the closed-form sum avoids iterative accumulation errors from floating-point drift. Note that the arithmetic series always diverges for d ≠ 0, so this calculator provides finite partial sums only. Remember that real-world indexed sequences (e.g., payment schedules) often start at index 0 or 1 - this tool uses 1-based indexing.

Formulas

The general (n-th) term of an arithmetic sequence is computed by the linear relation:

a_n = a₁ + (n − 1) ⋅ d

The partial sum of the first n terms uses Gauss's closed-form expression:

S_n = n2 ⋅ (2a₁ + (n − 1) ⋅ d)

Equivalently, when the last term a_n is known:

S_n = n ⋅ (a₁ + a_n)2

Where: a₁ = first term of the sequence, d = common difference (a_n+1 − a_n), n = number of terms (positive integer), a_n = the n-th (last) term, S_n = sum of the first n terms. The sum formula is quadratic in n: S_n = d2n² + (a₁ − d2)n, confirming that the graph of S_n vs. n is a parabola.

Reference Data

Property	Formula / Value	Notes
General Term	a_n = a₁ + (n − 1)d	Direct O(1) computation
Partial Sum (form 1)	S_n = n2(2a₁ + (n − 1)d)	Gauss formula using a₁ and d
Partial Sum (form 2)	S_n = n(a₁ + a_n)2	Useful when last term is known
Common Difference	d = a_n+1 − a_n	Constant for all consecutive pairs
Monotonicity	d > 0 → increasing	d < 0 → decreasing
Constant Sequence	d = 0	All terms equal a₁
Arithmetic Mean	a = a₁ + a_n2	Equals the average of first and last term
Sum of first 100 naturals	5050	a₁=1, d=1, n=100
Sum of first n odd numbers	n²	a₁=1, d=2
Sum of first n even numbers	n(n+1)	a₁=2, d=2
Convergence of Series	Diverges for d ≠ 0	Only converges trivially when a₁=0 and d=0
Number of terms from range	n = a_n − a₁d + 1	Only valid when d ≠ 0
Relation to quadratic	S_n is quadratic in n	S_n = d2n² + (a₁ − d2)n
Finite differences	Δa_n = d (constant)	First differences are constant; second differences are 0
Interpolation property	a_n = a_n−1 + a_n+12	Each term is the arithmetic mean of its neighbors
Gauss anecdote	1 + 2 + … + 100 = 5050	Attributed to young Carl Friedrich Gauss, c. 1786

Frequently Asked Questions

When d < 0, the sequence is strictly decreasing. Terms will eventually cross zero and become negative if a₁ > 0. The partial sum S_n initially increases, reaches a maximum near term index n ≈ 1 − 2a₁d, and then decreases as negative terms dominate.

The calculator caps at 1000 terms for the displayed sequence table to prevent excessive DOM rendering. The n-th term and partial sum formulas are O(1) and work for any n up to 10⁹ without performance issues, since they use closed-form expressions rather than iteration.

Yes. IEEE 754 double-precision floats have approximately 15-17 significant digits. For n > 10⁸ with non-integer d, the product n ⋅ d may lose precision. This calculator displays results to 10 significant figures maximum. For critical financial computations, use arbitrary-precision libraries.

If you know a_p and a_q (the p-th and q-th terms), compute d = a_q − a_pq − p. Then recover a₁ = a_p − (p − 1)d.

No. An infinite arithmetic series ∞∑n=1 a_n diverges for any d ≠ 0. Even when d = 0, the series diverges unless a₁ = 0. This is because the necessary condition for convergence (a_n → 0) fails when terms grow linearly.

An arithmetic sequence is the discrete analog of a linear function f(x) = d ⋅ x + (a₁ − d). The slope equals d, and the y-intercept equals a₁ − d. Plotting a_n vs. n always yields collinear points.