Mathematics · Precalculus
Arithmetic Series Calculator
Calculate the sum of the first n terms of an arithmetic sequence.
Inputs and results stay in this browser. Change one value at a time to explore the relationship.
Calculation steps
- Last term = 5 + (10−1)×3 = 32.
- Average first and last = (5+32)÷2 = 18.5.
- Sum = 10 × 18.5 = 185.
Understand Arithmetic series
One idea, three depths
Choose how deeply to explain Arithmetic series
Arithmetic series: Calculate the sum of the first n terms of an arithmetic sequence.
Age 5Explain it to a 5-year-oldStart with a picture
Imagine using Arithmetic series to answer this question: calculate the sum of the first n terms of an arithmetic sequence? Enter First term, Common difference, Number of terms; the calculator shows Series sum. For example: The first 10 terms starting at 5 with difference 3 sum to 185. The answer tells you Series sum.
Age 15Explain it to a 15-year-oldConnect it to the formula
Pairing terms from opposite ends gives repeated pairs with the same sum, producing the compact series formula. The rule is Sₙ = n/2[2a₁+(n−1)d]. Its input values are First term, Common difference, Number of terms, and the main result is Series sum. For example: The first 10 terms starting at 5 with difference 3 sum to 185.
CollegeExplain it at college levelState the model precisely
This calculator evaluates the stated arithmetic series relation over the valid mixed integer and real-number domain stated below. The implemented relation is Sₙ = n/2[2a₁+(n−1)d], evaluated from First term, Common difference, Number of terms to produce Series sum. Pairing terms from opposite ends gives repeated pairs with the same sum, producing the compact series formula. A sequence lists terms; a series adds them.
Inputs and valid domain
- First term must be a finite real number.
- Common difference must be a finite real number.
- Number of terms must be an integer, at least 1.
Important boundary: A sequence lists terms; a series adds them.
The formula
Sₙ = n/2[2a₁+(n−1)d]
How the calculator works through it
It substitutes First term, Common difference, Number of terms into the formula and exposes every numerical step above. The main output is Series sum, accompanied by Last included term.
Read the result correctly
The Series sum is the direct answer to “calculate the sum of the first n terms of an arithmetic sequence.” Read it with the units shown beside the inputs; a sign, angle, percentage or rate changes what the number means.
A worked check
The first 10 terms starting at 5 with difference 3 sum to 185.
Where this model stops being reliable
A sequence lists terms; a series adds them.
Learn it by changing one value
Begin with the worked example, then change one value while keeping the others fixed. Compare the new result and calculation steps to identify which part of the formula changed.
Dictionary terms behind this calculator
Before studying the codeWhat you should know firstUse the calculator immediately, or check the foundations before reading the implementation.
These foundations help you understand why Arithmetic series works. They never block the calculator, and “optional” means useful context rather than a hidden requirement.
Hard requirements
- Reading formulas and substituting values
Arithmetic series uses Sₙ = n/2[2a₁+(n−1)d]. You need to recognise what each side represents before substituting the stated inputs or rearranging the relationship.
Review this foundation about 4 min
Strong support
- Functions, domains and ranges
Domain and range language helps you identify which Arithmetic series inputs are valid and how the output behaves.
Review this foundation about 6 min
Optional enrichment
- Exponential growth and decay
Exponential models provide a useful extension when Arithmetic series is applied to multiplicative change.
Review this foundation about 6 min
Mathematics → algorithm → program
Implement this calculation in code
These are direct reference implementations of the calculator's principal relationship and first output. They run locally and include a small known-answer check where the language supports it.
Algorithm
- Read First term, Common difference, Number of terms.
- Evaluate the principal relationship: Sₙ = n/2[2a₁+(n−1)d].
- Return Series sum and check the domain conditions described above.
Python
from math import *
def arithmetic_series(a, b, n) -> float:
return ((n / 2.0) * ((2.0 * a) + ((n - 1.0) * b)))
assert abs(arithmetic_series(5, 3, 10) - 185) < 1e-6 * max(1.0, abs(185))
C
#include <assert.h>
#include <math.h>
double arithmetic_series(double a, double b, double n) {
return ((n / 2.0) * ((2.0 * a) + ((n - 1.0) * b)));
}
int main(void) {
const double expected = 185;
const double actual = arithmetic_series(5, 3, 10);
assert(fabs(actual - expected) < 1e-6 * fmax(1.0, fabs(expected)));
}
C++
#include <cassert>
#include <cmath>
#include <numbers>
double arithmetic_series(double a, double b, double n) {
return ((n / 2.0) * ((2.0 * a) + ((n - 1.0) * b)));
}
int main() {
constexpr double expected = 185;
const double actual = arithmetic_series(5, 3, 10);
assert(std::fabs(actual - expected) < 1e-6 * std::fmax(1.0, std::fabs(expected)));
}
Linux x86-64 assembly
x86-64 NASM · System V ABI · Linux · SSE2 with libm where required
; double arithmetic_series(double a, double b, double n)
; Linux x86-64 NASM · System V ABI · first eight doubles in xmm0–xmm7
global arithmetic_series
section .text
arithmetic_series:
push rbp
mov rbp, rsp
sub rsp, 96
movsd [rbp-8], xmm0
movsd [rbp-16], xmm1
movsd [rbp-24], xmm2
mov rax, 0x4000000000000000
movq xmm0, rax
movsd [rbp-48], xmm0
movsd xmm0, [rbp-24]
divsd xmm0, [rbp-48]
movsd [rbp-40], xmm0
mov rax, 0x4000000000000000
movq xmm0, rax
movsd [rbp-72], xmm0
movsd xmm0, [rbp-72]
mulsd xmm0, [rbp-8]
movsd [rbp-64], xmm0
mov rax, 0x3ff0000000000000
movq xmm0, rax
movsd [rbp-96], xmm0
movsd xmm0, [rbp-24]
subsd xmm0, [rbp-96]
movsd [rbp-88], xmm0
movsd xmm0, [rbp-88]
mulsd xmm0, [rbp-16]
movsd [rbp-80], xmm0
movsd xmm0, [rbp-64]
addsd xmm0, [rbp-80]
movsd [rbp-56], xmm0
movsd xmm0, [rbp-40]
mulsd xmm0, [rbp-56]
movsd [rbp-32], xmm0
movsd xmm0, [rbp-32]
leave
ret
MATLAB
function result = arithmetic_series(a, b, n)
result = ((n / 2.0) * ((2.0 * a) + ((n - 1.0) * b)));
end
Wolfram Language
ClearAll[mwCalculate];
mwCalculate[a_, b_, n_] := ((n / 2.0) * ((2.0 * a) + ((n - 1.0) * b)));
Continue in mathematical software
The downloaded file includes your current inputs and first calculated result. It is created locally.
Floating-point answers can differ slightly by language, compiler and processor. Compare within a suitable tolerance rather than assuming every decimal representation will be identical.
Supporting sourcesAcademic referencesPrimary standards, textbooks and complete citations
Standards, reading and academic references
Use the calculator as the worked interaction, then consult the primary standards and academic textbooks listed below. MW SysArc links to the original sources; the explanation on this page is original and does not reproduce them.
Algebra and Trigonometry 2e
Read the related free OpenStax mathematics chaptersCite this book
- APA 7
- Abramson, J. (2021). Algebra and trigonometry 2e. OpenStax. https://openstax.org/books/algebra-and-trigonometry-2e/pages/1-introduction-to-prerequisites
- MLA 9
- Abramson, Jay. Algebra and Trigonometry 2e. OpenStax, 2021, https://openstax.org/books/algebra-and-trigonometry-2e/pages/1-introduction-to-prerequisites.
- Chicago author-date
- Abramson, Jay. 2021. Algebra and Trigonometry 2e. Houston, TX: OpenStax. https://openstax.org/books/algebra-and-trigonometry-2e/pages/1-introduction-to-prerequisites.
OpenStax entries are free to read online. Follow the licence shown on each linked source before redistributing or adapting its content.
Reuse the page responsiblyCite this pageAPA, MLA, Chicago, Harvard, BibTeX and RIS
These formats cite this calculator page itself. They are separate from the academic references above, which support the mathematical method and terminology.
APA 7
MW SysArc. (2026, July 21). Arithmetic Series Calculator. MW SysArc Tools. https://math.mwsysarc.com/precalculus/arithmetic-series-calculator
MLA 9
MW SysArc. “Arithmetic Series Calculator.” MW SysArc Tools, 21 July 2026, https://math.mwsysarc.com/precalculus/arithmetic-series-calculator. Accessed 31 Aug. 2026.
Chicago 17
MW SysArc. “Arithmetic Series Calculator.” MW SysArc Tools. Published July 21, 2026. Accessed August 31, 2026. https://math.mwsysarc.com/precalculus/arithmetic-series-calculator.
Harvard
MW SysArc (2026) ‘Arithmetic Series Calculator’, MW SysArc Tools. Published 21 July 2026. Available at: https://math.mwsysarc.com/precalculus/arithmetic-series-calculator (Accessed: 31 August 2026).
BibTeX and RIS records
BibTeX
@misc{mwsysarc_arithmetic_series_2026,
author = {{MW SysArc}},
title = {Arithmetic Series Calculator},
howpublished = {MW SysArc Tools},
year = {2026},
url = {https://math.mwsysarc.com/precalculus/arithmetic-series-calculator},
note = {Published July 21, 2026; accessed August 31, 2026}
}RIS
TY - ELEC
AU - MW SysArc
TI - Arithmetic Series Calculator
T2 - MW SysArc Tools
PY - 2026
DA - 2026-07-21
Y2 - 2026-08-31
UR - https://math.mwsysarc.com/precalculus/arithmetic-series-calculator
N1 - Published July 21, 2026
ER -Clear answers
Frequently asked questions
What does the Arithmetic series do?
Calculate the sum of the first n terms of an arithmetic sequence.
How does the Arithmetic series work?
The calculator applies Sₙ = n/2[2a₁+(n−1)d]. Pairing terms from opposite ends gives repeated pairs with the same sum, producing the compact series formula.
What can I learn from the Arithmetic series?
It connects the mathematical rule to your chosen numbers and shows each calculation step. Change one input at a time to see how the result responds.
Does MW SysArc receive or store what I enter?
No. The calculation runs locally in your browser. MW SysArc does not receive or store your calculation inputs.
How should I use the result?
Use the steps to understand the method, then verify important school or professional work using the notation and rounding rules required in your setting.
Last reviewed . Calculations tested .