Arithmetic Sequence nth Term & Series Sum Calculator

Find a_n = a₁ + (n − 1)d and Gauss series sum S_n = n(a₁ + a_n)/2.

Partial Series Sum (S_n)
860
nth Term (a_n) 81
Sequence Mean ((a₁ + a_n)/2) 43
Next Term (a_{n+1}) 85
Inputs
First term of the sequence (a₁).
Common difference between consecutive terms (d).
Term index / number of terms (n).
Results & Graph
Partial Series Sum (S_n)
860
nth Term (a_n)
81
Sequence Mean ((a₁ + a_n)/2)
43
Next Term (a_{n+1})
85

Partial Series Sum (S_n)

Graph 1 • First Term A1 Sensitivity Recalculates the domain equation across 50% to 150% of the primary input while holding other parameters constant.
0 491.4 982.8 50%: 810 810 50% 75%: 835 835 75% 100%: 860 860 100% 125%: 885 885 125% 150%: 910 910 150%
Graph 2 • Primary & Supporting Metric Comparison Compares the primary output alongside supporting domain benchmarks.
Partial Series Sum (S_n) 860 nth Term (a_n) 81 Sequence Mean ((a₁ + a_n)/2) 43 Next Term (a_{n+1}) 85
Graph 3 • Exact Output Table & Units Displays each calculated output in its exact domain unit.
Partial Series Sum (S_n)860nth Term (a_n)81Sequence Mean ((a₁ + a_n)/2)43Next Term (a_{n+1})85
Summary: For a₁ = 5 and d = 4, the 20th term is a_20 = 81 and the sum of the first 20 terms is S_20 = 860.
Summary: For a₁ = 5 and d = 4, the 20th term is a_20 = 81 and the sum of the first 20 terms is S_20 = 860.

Formula & Mathematical Methodology: Arithmetic Series Mathematical Identity

an = a1 + (n − 1)d
Sn = (n / 2) × (2a1 + (n − 1)d)
a_n = a_1 + (n − 1)d; S_n = (n / 2) × (2a_1 + (n − 1)d)
  • firstTermA1: First Term A1 (numeric parameter)
  • commonDiffD: Common Diff D (numeric parameter)
  • termCountN: Term Count N (numeric parameter)

Rounding: All calculations run in double-precision IEEE-754 arithmetic; displayed outputs round to up to 4 decimal places.

Worked Example: Worked Mathematical Verification: Arithmetic Series

Find a_n = a₁ + (n − 1)d and Gauss series sum S_n = n(a₁ + a_n)/2.

  1. Verify the domain constraints of the input parameters.
  2. Apply the governing mathematical identity: a_n = a_1 + (n − 1)d; S_n = (n / 2) × (2a_1 + (n − 1)d).
  3. Cross-check the primary value against the supporting invariants and coordinate graph.
Result: Evaluates Arithmetic Series via a_n = a_1 + (n − 1)d; S_n = (n / 2) × (2a_1 + (n − 1)d).

Frequently Asked Questions

What mathematical formula does the Arithmetic Sequence nth Term & Series Sum Calculator use?

It directly evaluates a_n = a_1 + (n − 1)d; S_n = (n / 2) × (2a_1 + (n − 1)d) in double-precision floating-point arithmetic and updates all supporting invariants and graphs live.

How can I compare different parameter sets?

Click Compare in the results toolbar to pin up to three parameter combinations side by side.

Can I export the mathematical report or share a link?

Yes. Use the Share Link, Copy Summary, or Markdown/JSON export buttons in the results panel.

Mathematics & Statistics • Sequences & Series Guide

How to Use the Arithmetic Sequence nth Term & Series Sum Calculator: Practical Guide & Visual Analysis

Calculate the nth term a_n, partial sum S_n, and arithmetic mean of an arithmetic progression. Includes textbook mathematical notation, step-by-step verification, and three custom SVG visualizations. Whether you are researching arithmetic sequence nth term & series sum calculator, arithmetic series, arithmetic sequence sum, sequences & series calculator, this interactive tool helps you calculate the nth term a_n, partial sum S_n, and arithmetic mean of an arithmetic progression.

Interpreting the 3 Custom Visual Graphs & Key Outputs

As you adjust First Term A1, Common Diff D, Term Count N, the calculator immediately computes Primary exact or numerical solution, Supporting identities & invariants, 3-Panel mathematical SVG plots and updates all three domain-specific SVG charts in real time. Instead of relying on a single static figure, you can inspect the primary domain diagram, component breakdown, and multi-point trajectory curve to see how changing your inputs shifts the final result. Use the Compare (up to 3 scenarios) button to evaluate side-by-side variations before making a decision.

Real-World Benchmark & Planning Tips (Universal Mathematical & Statistical Standard (ISO 80000-2))

In our benchmark scenario (Worked Mathematical Verification: Arithmetic Series), Evaluates Arithmetic Series via a_n = a_1 + (n − 1)d; S_n = (n / 2) × (2a_1 + (n − 1)d). Keep in mind that uses standard Euclidean geometry, real/complex algebra, and NIST statistical definitions. For related calculations in Mathematics & Statistics, explore Scientific Calculator, Statistics & Standard Deviation Calculator, Geometry Calculator (Right Triangle, Circle & Cylinder).