Calculate the sum of an arithmetic sequence using S = n/2·(a₁ + aₙ). Enter first term, common difference, and number of terms. Shows sum, last term, average, partial sums, term table, and growth bars.
The sum of a linear (arithmetic) number sequence is one of the most famous results in mathematics. Legend has it that young Carl Friedrich Gauss astounded his teacher by instantly computing 1 + 2 + 3 + … + 100 = 5,050 using the formula S = n/2 · (first + last). This elegant formula works for any arithmetic sequence—a sequence where each term differs from the previous by a constant amount called the common difference.
An arithmetic sequence is defined by three parameters: the first term a₁, the common difference d, and the number of terms n. From these, you can compute the last term aₙ = a₁ + (n−1)·d, and the sum S = n/2 · (a₁ + aₙ) or equivalently S = n/2 · (2a₁ + (n−1)·d). The average term is simply S/n = (a₁ + aₙ)/2.
Arithmetic sequences appear everywhere: consecutive integers, even numbers, odd numbers, salary increments, depreciation schedules, seating arrangements in amphitheaters, and stacked objects. In finance, fixed periodic payments form arithmetic sequences. In computer science, many loop analyses involve summing arithmetic progressions.
This calculator computes the sum, last term, and average, then displays a table of individual terms with their partial sums and a visual growth bar chart showing how the cumulative sum evolves. Use the presets for classic examples like summing 1 to 100, or enter your own values.
Sum of Linear Number Sequence Calculator helps you solve sum of linear number sequence problems quickly while keeping each step transparent. Instead of redoing long algebra by hand, you can enter First Term (a₁), Common Difference (d), Number of Terms (n) once and immediately inspect Sum (S), Last Term (aₙ), Number of Terms to validate your work.
This is useful for homework checks, classroom examples, and practical what-if analysis. You keep the conceptual understanding while reducing arithmetic mistakes in multi-step calculations.
aₙ = a₁ + (n−1)·d. Sum S = n/2 · (a₁ + aₙ) = n/2 · (2a₁ + (n−1)·d). Average = S/n = (a₁ + aₙ)/2.
Result: Sum (S) shown by the calculator
Using the preset "1+2+…+100", the calculator evaluates the sum of linear number sequence setup, applies the selected algebra rules, and reports Sum (S) with supporting checks so you can verify each transformation.
This calculator takes First Term (a₁), Common Difference (d), Number of Terms (n), Terms to Show in Table and applies the relevant sum of linear number sequence relationships from your chosen method. It returns both final and intermediate values so you can audit the process instead of treating it as a black box.
Start with the primary output, then use Sum (S), Last Term (aₙ), Number of Terms, Average Term to confirm signs, magnitude, and internal consistency. If anything looks off, change one input and compare the updated outputs to isolate the issue quickly.
A strong workflow is manual solve first, calculator verify second. Repeating that loop improves speed and accuracy because you learn to spot common setup errors before they cost points on multi-step algebra problems.
An arithmetic sequence (or arithmetic progression) is a sequence where each term equals the previous term plus a fixed constant called the common difference d. Examples: 2, 5, 8, 11 (d=3) or 20, 15, 10, 5 (d=−5).
Gauss paired the numbers: 1+100=101, 2+99=101, 3+98=101, and so on. With 50 such pairs, the sum is 50×101=5,050. This is the formula S=n/2·(a₁+aₙ).
If d = 0, every term equals a₁, so the sum is simply n × a₁. The sequence is constant.
Yes. A negative d creates a decreasing sequence. The sum formula works the same way; the sum may end up negative if the sequence goes far enough below zero.
A sequence is the ordered list of terms (e.g., 1, 2, 3, …). A series is the sum of terms in a sequence (e.g., 1 + 2 + 3 + … = S). This calculator computes the series (sum) of an arithmetic sequence.
In an arithmetic sequence, you add a constant d each time. In a geometric sequence, you multiply by a constant ratio r. Their sum formulas are different: arithmetic uses S=n/2·(a₁+aₙ), geometric uses S=a₁·(1−rⁿ)/(1−r).