A=P\left(1+\frac{r}{n}\right)^{nt}Variables
- A: final amount
- P: principal
- r: annual rate as a decimal
- n: compounding periods per year
- t: years
How to use this formula
Calculates a balance with interest compounded a fixed number of times per year.
Important notes
- The stated rate must be converted from percent to decimal.
- This formula assumes a constant rate and no additional deposits or withdrawals.
Quick example
$1,000 at 5% compounded monthly for one year grows to about $1,051.16.
Applicability, worked calculation, and verification
Domain and applicability
Assumes a fixed nominal annual rate, regular compounding, no deposits or withdrawals, and consistent time units for n and t.
Units
- P and A use the same currency
- r is a decimal rate
- t is measured in years when r is annual
Assumptions and domain checks
- The stated rate must be converted from percent to decimal.
- For the Compound Interest Formula, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.
- Map every symbol to the modeled quantity and verify units, domain restrictions, and simplifying assumptions.
Do not use this formula when
- Do not use Compound Interest Formula when its variable definitions, domain restrictions, or structural assumptions differ from the problem.
Boundary and special cases
- For Compound Interest Formula, check zero, negative, and extreme input values before relying on the result.
- When using Compound Interest Formula, confirm that denominators, radicals, logarithms, and domain restrictions remain valid for the chosen values.
Equivalent and alternative forms
- Keep the canonical LaTeX form A=P\left(1+\frac{r}{n}\right)^{nt} for copying; rearrange only after preserving equivalence and domain restrictions.
Worked example
Invest $1,000 at a nominal annual rate of 5%, compounded monthly for 2 years: A = 1000(1 + 0.05/12)²⁴ ≈ $1,104.94.
- Set P = 1000, r = 0.05, n = 12, and t = 2.
- Compute the periodic rate: r/n = 0.05/12.
- Compute the number of periods: nt = 12 × 2 = 24.
- Evaluate 1000(1 + 0.05/12)²⁴ and round currency at the end.
Independent verification
The balance exceeds the principal and is slightly above the annual-compounding result, which is consistent with positive monthly compounding at the same nominal rate.
Common mistakes
- When copying Compound Interest Formula, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.
- Verify the result of Compound Interest Formula with a known case, inverse operation, dimensional check, or independent calculation before publishing it.
Continue the workflow
Use Compound Interest Formula in your own work
- Check the domainMatch the variables and assumptions to the problem before substituting values.
- Copy the exact notationPreserve grouping, signs, and exponents in
A=P\left(1+\frac{r}{n}\right)^{nt}. - Edit or convertOpen the expression in the LaTeX editor, then export it for your document or web page.
Review and verification
Last reviewed: 2026-07-26
Review method: Reviewed the notation, variable definitions, applicability conditions, worked workflow, and verification method for Compound Interest Formula.
Verified against: OpenStax Algebra and Trigonometry; NIST Digital Library of Mathematical Functions
Automated quality check: Passed the core formula indexing gate.
Formula references
- Algebra and Trigonometry 2eOpenStax, Rice University — Reviewed algebra, functions, sequences, trigonometry, and analytic geometry definitions and examples.
- Digital Library of Mathematical FunctionsNational Institute of Standards and Technology — Definitions, notation, identities, and reference material for mathematical functions.
Frequently asked questions
What is the Compound Interest Formula used for?
Calculates a balance with interest compounded a fixed number of times per year.
Can I copy this formula as LaTeX?
Yes. Copy A=P\left(1+\frac{r}{n}\right)^{nt} or open it in the LaTeX editor.
What should I check before using it?
Confirm that each variable, unit, domain restriction, and assumption matches the problem.