2026-09-29 8 min

Nth Term Calculator

Find any term of an arithmetic or geometric sequence: pick the mode, enter a1, the step, and n.

Calculator

Nth Term Inputs

First term (a1)

Step (d or r)

Position (n)

Sequence mode

Results update live as you type

Status:a(10) = 32 [arith]mode = arith

Nth Term Summary

Target term a(n)32

ModeArithmetic

Setupa1=5, step=3, n=10

Term list

5, 8, 11, 14, 17, 20, 23, 26, 29, 32

Preview shows the first 10 terms under the active mode.

Results

Optimal

Target Term a(n)

Under active mode

Mode

Arithmetic

Growth law

First Term

Step Used

d or r

Position n

Index

Mode-selected closed form with finite guards; preview list reveals wrong-mode picks.

Visualizations

Neighborhood terms

5a1

8a2

11a3

14a4

17a5

20a6

23a7

26a8

Term trajectory

Position in cap

n / 100010

a(n) vs a1

a(n)32

Formula

One page, two laws: add-steps for lines, multiply-steps for exponentials — the mode switch picks the engine.

Arithmetic: a(n) = a1 + (n-1)d · Geometric: a(n) = a1·r^(n-1)

Alternative Form

Mode decides: constant gaps vs constant ratios

Variables

Symbol	Name	Unit	Description
a1	First term	number	Seed of either mode
d / r	Step	number	Difference (arith) or ratio (geo)
n	Position	count	Target index >= 1
a(n)	Target term	number	Value under the chosen mode

Explanation

Identify the shape first (gaps vs ratios), then fire the matching closed form.

Derivation

Arithmetic by repeated addition; geometric by repeated multiplication — both proved by induction on the step count.

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 target.

Inputs: mode, a1, step, n

mode:arith

a1:5

step:3

n:10

Result: mode=arith, n=10

Step 2

Fire the mode-correct closed form.

arith: a1+(n-1)d · geo: a1·r^(n-1)

Intermediate: 5+(10-1)×3

Result: a(10) = 32

Step 3

Preview the shape for a mode sanity check.

First terms listed

Intermediate: 5, 8, 11, 14, 17, 20, 23, 26, 29, 32

Result: Shape reads arith

Step 4

Confirm flats are legitimate.

d=0 / r=1 repeat a1

Intermediate: step = 3

Result: a(10) = 32

Step 5

Cross-verify with a neighbor.

Recompute a(2)

Intermediate: a(2) = 8

Result: Neighbors consistent

Step 6

Publish with the mode attached.

a(n) + mode label

Intermediate: 32 [arith]

Result: Verified a(10) = 32

Final Answer

Worked Examples

See how the calculator works with different scenarios.

Scenario

a1=5, d=3, n=10.

Inputs

mode:arith

a1:5

step:3

n:10

Calculation

a(10) = 5+9×3 = 32

Result

a(10) = 32

Interpretation

Nine adds of 3.

Understanding Your Result

Your Result: a(n) under your mode — see the target card.

Typical range: n caps at 1000; geo overflow guarded; arith d = 0 and geo r = 1 both valid flats.

What This Means

The value at exactly position n under the selected growth law.

Why It Matters

Homework asks 'find a(20)' constantly — mode clarity plus the right formula ends the confusion.

Professional Interpretation

The preview list exposes the wrong-mode mistake instantly: lines stay straight, exponentials bend.

Recommendations

•Pick mode before numbers
•Confirm gaps vs ratios
•Quote a(n) with mode
•Keep n integer ≥ 1
•Verify neighbors match mode

Educational Content

Introduction

Stop guessing add-vs-multiply: declare the family, then let the correct a(n) fire.

Concept Explanation

Two engines, one dashboard — shape diagnosis drives formula choice.

Applications

▸Homework nth-term drills

▸Salary-ladder vs growth forecasts

▸Depreciation vs compounding

▸Seating vs bacterial counts

▸Quiz answer checking

▸Pattern continuation

Benefits

•Dual-mode comparison
•Shape preview
•Flat cases valid
•Overflow guards
•Steps per mode

Limitations

•Two families only
•n <= 1000
•Constant step assumed

Best Practices

•Diagnose shape first
•State mode in answers
•Check two neighbors
•Keep n whole
•Re-run on mode flip

Common Misconceptions

Myth: One formula covers both families.

Truth: Addition and multiplication need separate closed forms.

Myth: Step means the same in both modes.

Truth: Difference adds; ratio multiplies — units of thinking differ.

Myth: Flat outputs are errors.

Truth: d=0 and r=1 flats are legitimate.

Myth: n starts at 0.

Truth: Here a1 is position 1.

Practical Tips

•Diagnose gaps vs ratios first
•Flip the mode to compare
•Keep n a positive integer
•Read flats as valid
•Verify with neighbors
•Quote mode alongside a(n)
•Watch geo overflow early

Common Mistakes

Firing arithmetic on geometric data

Ratios constant → geometric; gaps constant → arithmetic.

Using d as a multiplier

In geo mode the step multiplies (r); in arith it adds (d).

Exponent n instead of n-1 (geo)

First term takes zero jumps.

Forgetting 1-based indexing

a(1) is the seed, not a(0).

Ignoring overflow flags

Shorten n or shrink |r| when warned.

Assumptions

This calculator makes the following assumptions to provide estimates:

Mode governs: One law per run.

Constant step: d or r fixed.

Whole index: n integer 1..1000.

Double math: Finite-guarded.

Local only: In-browser.

Limitations

Be aware of these limitations when using this calculator:

Two laws: Quadratic/harmonic need their pages.

No auto-detect: You choose the mode.

Finite preview: First 10 print.

Power ceiling: Geo overflow guarded.

Frequently Asked Questions

Answers to common questions about this calculator.

How do I find a(20) of 5, 8, 11, ...?

Arithmetic: a1=5, d=3 → a(20)=5+19×3=62.

How do I find a(10) of 3, 6, 12, ...?

Geometric: a1=3, r=2 → 3×2^9=1536.

Add or multiply?

Constant gaps → add (arithmetic). Constant ratios → multiply (geometric).

What is the difference formula?

a(n)=a1+(n-1)d.

What is the ratio formula?

a(n)=a1·r^(n-1).

Why two answers for one n?

Different laws extrapolate differently — mode is part of the question.

Flat sequences?

d=0 repeats a1; r=1 repeats a1. Both valid.

Which grows faster?

Geometric eventually dominates any arithmetic.

n = 0?

Out of scope; positions start at 1.

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.

nth term: Value at position n.
Arithmetic mode: a1+(n-1)d engine.
Geometric mode: a1·r^(n-1) engine.
Common difference: Arith step d.
Common ratio: Geo step r.
Mode: Selected growth family.
Seed: First term a1.
Shape diagnosis: Gaps-vs-ratios check.
Flat case: d=0 or r=1 repeats.

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.