The Arithmetic Sequence Calculator finds any term in an arithmetic sequence and computes the sum of the first n terms from the first term and common difference. Used in algebra, financial mathematics, and any application involving regularly spaced numerical progressions.
98
1,010
50.5
8
13
48
248
498
98
1,010
50.5
8
13
48
248
498
Every time you count by the same amount — 3, 6, 9, 12... or 100, 95, 90, 85... — you are working with an arithmetic sequence. The calculator for arithmetic sequences finds any specified term, lists the first n terms, and computes the partial sum, making sequence problems that would require dozens of manual calculations instant and error-free.
An arithmetic sequence has first term a₁ and common difference d (the constant amount added between consecutive terms). The key formulas:
nth term: aₙ = a₁ + (n−1)d
Sum of first n terms: Sₙ = n/2 × (a₁ + aₙ) = n/2 × (2a₁ + (n−1)d)
For the sequence 3, 7, 11, 15, ... (a₁ = 3, d = 4): the 20th term = 3 + 19 × 4 = 79; sum of first 20 terms = 20/2 × (3 + 79) = 10 × 82 = 820. The sum formula is derived by pairing the first and last terms: (a₁ + aₙ) + (a₂ + aₙ₋₁) + ... all equal the same value, and there are n/2 such pairs. This elegant result was famously used by young Gauss to instantly sum 1 + 2 + ... + 100 = 100/2 × 101 = 5,050. Use this online calculator to find any term or partial sum for any arithmetic sequence. The arithmetic series calculator focuses on the summation aspect.
A sequence is arithmetic if and only if the difference between consecutive terms is constant: d = a₂ − a₁ = a₃ − a₂ = ... = aₙ − aₙ₋₁. Testing whether a sequence is arithmetic: compute the first differences; if they are all equal, the sequence is arithmetic and d equals that constant difference. Mixed arithmetic sequences (where d changes at some point) are not arithmetic — they are piecewise arithmetic and must be treated as separate sequences.
Arithmetic sequences model any situation where a quantity changes by a fixed amount in each period:
The geometric sequence calculator handles the multiplicative counterpart where each term is multiplied by a constant ratio. The sequences and series calculators cover the complete toolkit.
When two non-consecutive terms of an arithmetic sequence are known, the first term and common difference can be recovered: if aₘ and aₙ are known (m ≠ n), then d = (aₙ − aₘ)/(n − m) and a₁ = aₘ − (m−1)d. Example: if the 5th term is 23 and the 13th term is 55: d = (55−23)/(13−5) = 32/8 = 4; a₁ = 23 − 4×4 = 7. Verify: a₅ = 7 + 4×4 = 23 ✓; a₁₃ = 7 + 12×4 = 55 ✓. This reverse-engineering capability makes arithmetic sequences useful for fitting linear models to discrete data.
Given first term $$a_1$$ and common difference $$d$$:
n-th Term Formula:
$$a_n = a_1 + (n-1)d$$
This is a linear function of $$n$$ with slope $$d$$ and intercept $$a_1 - d$$.
Partial Sum Formula:
$$S_n = \frac{n}{2}(a_1 + a_n) = \frac{n}{2}(2a_1 + (n-1)d)$$
This can be derived by writing the sum forward and backward, then adding:
$$S_n = a_1 + (a_1+d) + \cdots + a_n$$
$$S_n = a_n + (a_n-d) + \cdots + a_1$$
$$2S_n = n(a_1 + a_n)$$
Arithmetic Mean:
$$\bar{a} = \frac{S_n}{n} = \frac{a_1 + a_n}{2}$$
The mean of an arithmetic sequence is simply the average of the first and last terms, regardless of how many terms there are.
n-th Term (aₙ) is the value of the sequence at position $$n$$. For positive $$d$$, the sequence increases; for negative $$d$$, it decreases; for $$d = 0$$, all terms are equal to $$a_1$$.
Sum of First n Terms (Sₙ) grows quadratically with $$n$$ (since $$S_n = \frac{d}{2}n^2 + (a_1 - \frac{d}{2})n$$). This quadratic growth distinguishes arithmetic sums from geometric sums.
Arithmetic Mean is always the midpoint between $$a_1$$ and $$a_n$$. This property makes arithmetic sequences special — the mean of any symmetric subset equals the overall mean.
The reference terms (2nd, 3rd, 10th, 50th, 100th) give you a quick overview of how the sequence behaves across different scales.
Inputs
Results
The 20th term: a₂₀ = 3 + 19·5 = 98. Sum: S₂₀ = 20·(3+98)/2 = 20·50.5 = 1010. The arithmetic mean is 50.5, which is exactly (3+98)/2 — the midpoint between first and last terms.
Inputs
Results
With d = -7, the sequence decreases by 7 each step. The 15th term: a₁₅ = 100 + 14·(-7) = 100 - 98 = 2. Sum: S₁₅ = 15·(100+2)/2 = 15·51 = 765. Eventually the terms become negative (a₁₅ = 2, a₁₆ = -5).
The common difference d is the constant value added to each term to get the next term. If d > 0, the sequence increases; if d < 0, it decreases; if d = 0, all terms are identical. You can find d from any two consecutive terms: d = aₙ₊₁ - aₙ.
Write the sum forwards (S = a₁ + a₂ + ... + aₙ) and backwards (S = aₙ + aₙ₋₁ + ... + a₁). Add these: 2S = n·(a₁ + aₙ) because each pair sums to a₁ + aₙ. Therefore S = n·(a₁ + aₙ)/2. This is Gauss's method, discovered when he was seven years old.
Yes. The formulas work for any real number d. For example, d = 0.5 gives the sequence 1, 1.5, 2, 2.5, ... and d = 1/3 gives 0, 1/3, 2/3, 1, 4/3, .... The results may have many decimal places.
An arithmetic sequence aₙ = a₁ + (n-1)d is the discrete version of the linear function f(x) = dx + (a₁ - d). The common difference d corresponds to the slope. Plotting aₙ versus n gives points on a straight line.
If you know aₘ and aₙ (terms at positions m and n), then d = (aₙ - aₘ)/(n - m), and a₁ = aₘ - (m-1)d. For example, if a₃ = 10 and a₇ = 26, then d = (26-10)/(7-3) = 4, and a₁ = 10 - 2·4 = 2.
The arithmetic mean of the sequence's first n terms equals (a₁ + aₙ)/2 — the average of the first and last terms. This elegant property holds because the terms are symmetrically distributed around the center value. It's also why the sum formula works: S = n × mean.
How helpful was this calculator?
5.0/5 (1 rating)
