2026-09-29 9 min

Recursive Formula Calculator

Derive the recurrence a(n) = p·a(n-1) + q from parameters, show the closed form, and sketch the induction proof.

Calculator

Recurrence Definition Inputs

Base a(1)

Multiplier (p)

Addend (q)

Target (n)

Results update live as you type

Status:a(4) = 52base holds · forms agree

Recursive Formula Summary

a(n)52

Base holdsYes

Setupa1=3, p=2, q=4, n=4

Term list

3, 10, 24, 52

Preview iterates the rule 10 steps; the proof covers all n by induction.

Results

Optimal

a(n) via Rule

Iterated recurrence

Closed Form Value

Proved shortcut

Base Case Holds

Yes

n = 1 agreement

|Rule − Closed|

Should be ~0

Target n

Index

Base case, inductive substitution, and numeric rule-vs-closed agreement are all checked live.

Visualizations

Recurrence terms

3a1

10a2

24a3

52a4

Recurrence trajectory

Induction target

n / 10004

Rule value vs closed value

rule52

Formula

The recurrence is the definition; the claim is its consequence — induction bridges them step by step.

a(1) = a1, a(n) = p·a(n-1) + q

Alternative Form

Claim: p=1 → a1+(n-1)q; else a1·p^(n-1)+q(p^(n-1)-1)/(p-1)

Variables

Symbol	Name	Unit	Description
a1	Base case	number	Anchors the induction
p	Recurrence multiplier	number	Coefficient of a(n-1)
q	Nonhomogeneous term	number	Added each level
a(n)	Closed value	number	Proved formula at n

Explanation

Base case plus inductive step equals a proof: assume the claim at n−1, push through the rule, land on n.

Derivation

Base: claim holds at n=1 (p^0=1). Step: substitute the hypothesis for a(n-1) into p·a(n-1)+q and simplify to the claim at n.

Step-by-Step Calculation

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

Step 1

State the recurrence and base case.

a(1)=a1, a(n)=p·a(n-1)+q

a1:3

p:2

q:4

n:4

Result: Rule on a1=3

Step 2

Check the base case of the claim.

claim(1) = a1

Intermediate: closed(1) equals a1

Result: Base holds

Step 3

Assume the claim at n-1 (hypothesis).

IH: closed(n-1)

Intermediate: 3, 10, 24, 52

Result: Hypothesis staged

Step 4

Push the hypothesis through the rule.

p·IH + q → claim(n)

Intermediate: algebra: geometric tail extends by one power

Result: lands on 52

Step 5

Conclude the claim for all n by induction.

base + step ⇒ ∀n

Intermediate: a(4) = 52

Result: Proved for all n

Step 6

Audit numerically at your n.

|rule − closed| ≈ 0

Intermediate: 52 vs 52

Result: gap = 0

Final Answer

Worked Examples

See how the calculator works with different scenarios.

Scenario

a1=3, p=2, q=4, n=4.

Inputs

a1:3

p:2

q:4

n:4

Calculation

Iterate: 3,10,24,52 Closed: 3·8+4·7=52

Result

a(4) = 52

Interpretation

Rule and claim coincide.

Understanding Your Result

Your Result: Recurrence plus proved closed form for your p, q — see the a(n) card.

Typical range: n caps at 1000; p = 1 handled as its own arithmetic claim.

What This Means

Two views of one sequence: the step-by-step rule and the jump-to-n formula it implies.

Why It Matters

Publishing the rule without the closed form forces loops; proving it once unlocks every n forever.

Professional Interpretation

The page checks base, hypothesis-substitution, and numeric agreement — the full induction skeleton, computed.

Recommendations

•State base case first
•Isolate p = 1 cleanly
•Show the substitution algebra
•Confirm numbers match
•Quote rule + closed form together

Educational Content

Introduction

Write the recurrence, conjecture its shortcut, then let induction certify it — all around your numbers.

Concept Explanation

Induction mechanics made concrete: base holds, step preserves, therefore all n hold.

Applications

▸Textbook induction drills

▸Algorithm recurrence proofs

▸Savings-plan derivations

▸Dosing-schedule justifications

▸Homework rule-to-form tasks

▸Interview recurrence rounds

Benefits

•Rule + claim side by side
•Base/step proof sketch
•p = 1 branch clean
•Numeric agreement audit
•Fixed point context

Limitations

•First-order linear only
•n <= 1000
•Proof sketch, not formal logic

Best Practices

•Write a(1) explicitly
•Show hypothesis use
•Simplify to target form
•Test n = 1, 2
•Keep p, q constant

Common Misconceptions

Myth: Examples prove the formula.

Truth: Only base + general step prove all n; examples merely illustrate.

Myth: The closed form is a different sequence.

Truth: Induction shows it generates identical terms.

Myth: p = 1 needs division by zero.

Truth: It gets its own arithmetic claim — no division.

Myth: Induction runs forward only.

Truth: It chains: base triggers step 1→2→3→...

Practical Tips

•Anchor a(1) before anything
•Name p and q in the rule
•Substitute the hypothesis visibly
•Simplify toward the n-form
•Check n = 1 and n = 2 numerically
•Split p = 1 out early
•Publish rule with its proof

Common Mistakes

Skipping the base case

Verify n = 1 gives a1 on both sides first.

Assuming the claim at n instead of n-1

Assume at n-1, derive at n — direction matters.

Algebra slip in q-terms

Collect q·p^(n-2) + q carefully into the geometric tail.

Dividing by (p-1) when p = 1

Use the arithmetic claim there.

Testing only one numeric case

Check base plus agreement at your n.

Assumptions

This calculator makes the following assumptions to provide estimates:

Linear first-order: Rule shape fixed.

Stated base: a(1) = a1 given.

Whole n: n integer 1..1000.

Exact algebra: Symbolic step + numeric check.

Local only: Browser-side.

Limitations

Be aware of these limitations when using this calculator:

Proof sketch: Pedagogical outline, not machine-checked.

Order one: Higher orders out of scope.

Constant params: p, q fixed.

Finite demo: Preview lists 10 terms.

Frequently Asked Questions

Answers to common questions about this calculator.

What is a recursive formula?

A rule from predecessors, e.g. a(1)=3, a(n)=2·a(n-1)+4.

What is its closed form?

a(n)=3·2^(n-1)+4(2^(n-1)-1): unroll the +4 tail as a geometric sum.

How does induction prove it?

Check n=1; assume the form at n-1, feed it through the rule, simplify to the n-form.

Why p = 1 special?

The geometric tail becomes plain (n-1)q — separate arithmetic claim.

Do I need the proof to compute?

No, but it guarantees the shortcut for every n.

Fixed point role?

Centers the algebra: shifting by a* homogenizes the rule.

Example numbers?

a1=3,p=2,q=4,n=4 → 3,10,24,52; closed: 3·8+4·7=52.

How is this page different?

The recursive-sequence page iterates; this one derives, proves, then evaluates.

Negative p?

Proof still holds; terms oscillate.

Stored?

No.

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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.

Recurrence relation: Definition via predecessors.
Base case: Verified start of induction.
Inductive step: n-1 ⇒ n implication.
Inductive hypothesis: Assumed claim at n-1.
Closed form: Direct formula proved equivalent.
Nonhomogeneous: The +q driving term.
Fixed point: q/(1-p) self-loop value.
Telescoping sum: Cancellation inside the unrolling.
Base trigger: Case starting the domino chain.

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.