About

An arithmetic series is the summation of terms in an arithmetic progression where each consecutive term differs by a fixed value d, the common difference. Miscalculating even one parameter propagates errors through every dependent result - partial sums, term indices, convergence checks in numerical methods. This calculator solves all four core unknowns: the n-th term a_n, the partial sum S_n, the number of terms n, and the common difference d. It applies the closed-form Gauss formula S_n = n2 ⋅ (a₁ + a_n) rather than iterative addition, so results are exact regardless of series length.

The tool assumes a finite, non-degenerate progression (n ≥ 1). For d = 0 the series reduces to n ⋅ a₁. Negative values for d produce decreasing sequences. When solving for n, only positive integer solutions are accepted; non-integer results indicate the given parameters do not form a valid arithmetic series. Pro tip: in financial amortization schedules and depreciation tables, arithmetic progressions appear constantly. Verify your inputs against the source ledger before relying on computed sums.

Formulas

The primary formula for the n-th term of an arithmetic progression with first term a₁ and common difference d:

a_n = a₁ + (n − 1) ⋅ d

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

S_n = n2 ⋅ (2a₁ + (n − 1) ⋅ d)

Equivalently, when the last term a_n is known:

S_n = n2 ⋅ (a₁ + a_n)

Solving for the common difference when a₁, a_n, and n are known:

d = a_n − a₁n − 1

Solving for n when a₁, d, and a_n are known:

n = a_n − a₁d + 1

Where: a₁ = first term, d = common difference, n = number of terms, a_n = last (n-th) term, S_n = sum of the first n terms.

Reference Data

Property	Formula / Value	Notes
General term	a_n = a₁ + (n − 1)d	Explicit (closed-form) definition
Partial sum (form 1)	S_n = n2(2a₁ + (n − 1)d)	When a_n is unknown
Partial sum (form 2)	S_n = n2(a₁ + a_n)	Gauss's formula; when last term known
Arithmetic mean	S_nn = a₁ + a_n2	Always equals midpoint of first and last term
Common difference	d = a_n − a_n−1	Constant for all consecutive pairs
Number of terms	n = a_n − a₁d + 1	Must yield a positive integer
Sum of first n naturals	n(n + 1)2	a₁ = 1, d = 1
Sum of first n odd numbers	n²	a₁ = 1, d = 2
Sum of first n even numbers	n(n + 1)	a₁ = 2, d = 2
Monotonicity	Increasing if d > 0; Decreasing if d < 0	Constant if d = 0
Recursive form	a_n = a_n−1 + d	Inductive definition
Finite vs infinite	Finite AP has a well-defined sum; infinite AP diverges unless d = 0	No convergence for d ≠ 0
Relation to quadratic	S_n is a quadratic function of n	Parabolic growth in n
Middle term (odd n)	a_(n+1)/2 = a₁ + a_n2	Equals the arithmetic mean
AP test	2b = a + c	Three terms a, b, c are in AP iff this holds
Gauss anecdote	1 + 2 + … + 100 = 5050	Classic example; n = 100, d = 1

Frequently Asked Questions

When d = 0, every term equals a₁. The sum simplifies to S_n = n ⋅ a₁. The sequence is constant and the arithmetic mean equals a₁. This is a valid, non-degenerate case.

Yes. A negative d produces a strictly decreasing sequence. A fractional d (e.g., 0.5) is equally valid. The closed-form formulas work for all real-valued d without modification.

The formula n = (a_n − a₁) ÷ d + 1 requires d ≠ 0 and must yield a positive integer. If the result is fractional or negative, the given a₁, a_n, and d do not form a valid finite arithmetic progression.

The partial sum S_n = d2n² + (a₁ − d2)n is a quadratic in n. This means if you plot S against n, you get a parabola. The coefficient of n² is d2, so the parabola opens upward when d > 0 and downward when d < 0.

The calculator computes results for any positive integer n, but the full sequence listing is capped at 500 terms to prevent browser performance degradation. The sum and nth term are always computed exactly via the closed-form formula regardless of n.

Three numbers a, b, c are in arithmetic progression if and only if 2b = a + c. Equivalently, b − a = c − b. This calculator uses this identity internally when validating inputs.