2026-09-29 8 min

Partial Sum Calculator

Sum the first n terms in arithmetic or geometric mode: Sn = n(a1+an)/2 or a1(1-r^n)/(1-r), with average and checks.

Calculator

Partial Sum Inputs

First term (a1)

Step (d or r)

Terms to sum (n)

Sum mode

Results update live as you type

Status:S(10) = 100 [arith]avg = 10

Partial Sum Summary

Partial sum S(n)100

Average term10

Setuparith · a1=1, step=2, n=10

Term list

1, 3, 5, 7, 9, 11, 13, 15, 17, 19

Preview lists the first 10 terms; S(n) always covers all n.

Results

Optimal

Partial Sum S(n)

100

Mode-correct total

nth Term

Last included

Average Term

S(n)/n

Mode

Arithmetic

Active law

Check n × avg

100

Must equal S(n)

Count n

Terms

Mode-correct closed forms with the r=1 flat branch; n·avg cross-check runs live.

Visualizations

Summed terms

1a1

4a2

9a3

16a4

25a5

36a6

49a7

64a8

Accumulation path

Terms totaled

n / 100010

Sum vs nth term

sum100

Formula

Totals ride on the term law: averages for lines, telescoping fractions for exponentials.

Arith: S(n) = n(a1+an)/2 · Geo: a1(1-r^n)/(1-r)

Alternative Form

Flat geo: S(n) = n·a1 when r = 1

Variables

Symbol	Name	Unit	Description
a1	First term	number	Summed first
step	Step	number	d (arith) or r (geo)
n	Count	count	Terms totaled (>= 1)
S(n)	Partial sum	number	Total under active mode

Explanation

Every partial sum is count-times-average in disguise — geometric averaging just hides inside the fraction.

Derivation

Arithmetic via endpoint pairing; geometric via S − rS telescoping; r = 1 flat by repeated addition.

Step-by-Step Calculation

Follow each step to understand how the result is calculated from your inputs.

Step 1

Read the mode, seed, step, and count.

Inputs: mode, a1, step, n

mode:arith

a1:1

step:2

n:10

Result: mode=arith, n=10

Step 2

Find the last included term.

arith / geo nth forms

Intermediate: an = 19

Result: an = 19

Step 3

Apply the mode-correct total.

pairing or telescoping

Intermediate: closed form

Result: S(10) = 100

Step 4

Derive the average term.

avg = S(n)/n

Intermediate: 100/10

Result: avg = 10

Step 5

Cross-check n × avg against Sn.

n × avg = S(n)

Intermediate: 10×10

Result: check = 100

Step 6

Verify with the preview list.

Direct add of preview

Intermediate: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19

Result: Verified S(10) = 100

Final Answer

100

Worked Examples

See how the calculator works with different scenarios.

Scenario

a1=1, d=2, n=10.

Inputs

mode:arith

a1:1

step:2

n:10

Calculation

an=19, S=10·20/2=100

Result

S(10) = 100

Interpretation

Odds total a square.

Understanding Your Result

Your Result: S(n) under your mode — see the sum card, verified by n × average.

Typical range: n caps at 1000; geo |r|^n guarded; negative steps handled exactly.

What This Means

The running total after the first n terms of the chosen progression.

Why It Matters

Invoices, balances, and staged deliveries total as partial sums — mode-correct math prevents pricing errors.

Professional Interpretation

n × average must reproduce Sn in arithmetic mode — the page checks it every keystroke.

Recommendations

•Pick mode before totals
•Verify Sn = n × avg (arith)
•Isolate r = 1
•Quote S(n) with count
•Hand-check n = 1, 2

Educational Content

Introduction

Totals with a mode switch: lines average, exponentials telescope — one dashboard for both.

Concept Explanation

Partial sums inherit their family's proof: pairing vs multiply-and-subtract.

Applications

▸Staged-payment totals

▸Inventory build-up sums

▸Balance roll-ups

▸Tiered-revenue totals

▸Training-volume sums

▸Material takeoffs

Benefits

•Dual-mode totals
•Average bundled
•Flat branch clean
•Telescoping proof
•Accumulation chart

Limitations

•Two families
•n <= 1000
•Constant step

Best Practices

•State mode in reports
•Check n·avg
•Split r = 1
•Verify tiny n
•Cap far forecasts

Common Misconceptions

Myth: Partial sum means infinite sum.

Truth: Partial = first-n total; infinite is a limit question.

Myth: One sum formula fits all.

Truth: Arithmetic and geometric need different closed forms.

Myth: Average is only for statistics.

Truth: Sn/n is the average term — the pairing engine.

Myth: r = 1 needs the fraction.

Truth: It needs n·a1; the fraction divides by zero.

Practical Tips

•Choose mode before numbers
•Confirm Sn = n × average (arith)
•Split r = 1 out first
•Keep n a positive integer
•Hand-check S(1) = a1
•Quote S(n) with its count
•Watch last-term dominance (geo)

Common Mistakes

Summing with the wrong family

Diagnose gaps vs ratios before totaling.

Dropping /2 in arithmetic

Pairing gives n/2 pairs — keep it.

Dividing by (1-r) at r = 1

Switch to n·a1.

Reading Sn as average

Divide by n for the mean.

Fractional n

Floor to whole counts ≥ 1.

Assumptions

This calculator makes the following assumptions to provide estimates:

Mode governs: One law per run.

Constant step: d/r fixed.

Whole count: n integer 1..1000.

Double totals: Guarded arithmetic.

Local only: In-browser.

Limitations

Be aware of these limitations when using this calculator:

Finite partials: Infinite limits elsewhere.

Two families: Others need sibling pages.

Preview cap: First 10 print.

Power ceiling: Geo overflow guarded.

Frequently Asked Questions

Answers to common questions about this calculator.

What is S(10) of 1,3,5,...?

Arithmetic a1=1,d=2: an=19, S=10(1+19)/2=100.

What is S(8) of 1,2,4,...?

Geometric a1=1,r=2: (1-256)/(1-2)=255.

Which formula when?

Even gaps → arithmetic; even ratios → geometric.

Why n·avg check?

It re-derives Sn independently — mismatch means wrong inputs.

r = 1?

S(n)=n·a1.

Negative steps?

Handled: arith totals can peak; geo totals oscillate.

Partial vs infinite?

Partial stops at n; infinite asks the limit.

S(1)?

Always a1 — the fastest sanity check.

Huge n?

Capped at 1000 for finiteness.

Stored?

No.

Related Calculators

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Related Guides

Deepen your understanding with these educational resources.

Sequences vs series in one sketch

Terms, sums, and convergence at a glance.

Closed forms that skip the loop

Jump straight to term n without listing.

Growth regimes: linear vs exponential

Reading difference and ratio correctly.

Checking sums by pairing and ratio tricks

Endpoint pairing and the r-times-Sn proof.

Glossary

Technical terms used in mortgage calculations explained in plain language.

Partial sum: Total of first n terms.
Arithmetic total: n(a1+an)/2.
Geometric total: a1(1-r^n)/(1-r).
Average term: Sn/n.
Flat case: r=1 → n·a1.
Pairing: Endpoint-sum trick.
Telescoping: S−rS cancellation.
Count: n Chun.
Mode: Active family.

Authoritative References

Trusted sources and standards referenced by this calculator.

Wikipedia — Sequence Wikipedia — Series (mathematics) Khan Academy — Sequences Math is Fun — Sequences and Series Purplemath — Sequences and Series Wolfram MathWorld — Sequence Cuemath — Sequences and Series

Applicable Standards

Common Core HSF-BF.A — building sequences and seriesNCTM Number and Operations — patterns, functions, and series

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Page Information

Last Updated:

2026-09-29

Version:

2.0.0

References:

License:

Free to use for educational and professional purposes.