Sum of Linear Number Sequence Calculator
An arithmetic sequence increases (or decreases) by the same fixed amount each step. Enter the first term, common difference d and number of terms n to get the total sum and last term instantly.
Sₙ = n/2 × (first + last term)
- 1
Last term aₙ = a₁ + (n − 1) × d
1 + (10 − 1) × 2 = 19 - 2
First plus last term
1 + 19 = 20 - 3
Sum Sₙ = n ÷ 2 × (a₁ + aₙ)
10 ÷ 2 × 20 = 100
How does this calculator work?
Sum of an arithmetic sequence = n/2 × (first + last) = n/2 × (2a₁ + (n−1)d). Enter the first term, common difference d and number of terms n. Handles any real values including negatives. Result is exact with no rounding beyond standard floating-point precision.
Formula
How this is calculated
An arithmetic (linear) sequence is a list of numbers where consecutive terms differ by a fixed constant d called the common difference. For example 3, 6, 9, 12 has a₁ = 3 and d = 3. The n-th term is aₙ = a₁ + (n − 1)d.
The sum formula Sₙ = n/2 × (a₁ + aₙ) works because pairing the first and last terms, the second and second-to-last, and so on gives the same constant total each time — this is the insight attributed to Gauss as a child. Substituting aₙ = a₁ + (n − 1)d gives the fully expanded form Sₙ = n/2 × (2a₁ + (n − 1)d). Both forms are algebraically identical and exact with no approximation.
The formula handles any real-valued first term and common difference, including negative values and zero. A negative d produces a decreasing sequence; d = 0 makes every term equal to a₁ and the sum simplifies to n × a₁.
Frequently asked questions
A sequence where each term equals the previous one plus a constant d (the common difference). Examples: 2, 5, 8, 11 (d = 3); 10, 7, 4, 1 (d = −3). The terms form a straight line when plotted against their index — hence "linear".
Compute Sₙ (sum of the first n terms) and Sₘ₋₁ (sum of the first m−1 terms) separately using the calculator, then subtract: Sₙ − Sₘ₋₁.
The sequence is constant — every term equals a₁ — and the sum simplifies to n × a₁.
Also known as
TG we-Calculate Editorial Team. (2026). Sum of Linear Number Sequence Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/sum-of-linear-number-sequence-calculator
TG we-Calculate Editorial Team. "Sum of Linear Number Sequence Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/sum-of-linear-number-sequence-calculator.
TG we-Calculate Editorial Team, "Sum of Linear Number Sequence Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/sum-of-linear-number-sequence-calculator
@misc{wecalculate_sum_of_linear_number_sequence_calculator, title = {Sum of Linear Number Sequence Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/sum-of-linear-number-sequence-calculator}}, year = {2026}, note = {TG we-Calculate} }
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