ALG-14

Compound interest

A = P (1 + r/n)^{nt}. Principal P after t years.

Reading speed

ExponentialCompound interest

Governing equation

A=P\left(1+\dfrac r n\right)^{nt}

$t \in \left[0, 50\right]$

where

P: Principal (—)
r: Nominal rate (1/yr)
n: Compounds per year (1/yr)
t: Years (yr)
A: Amount (—)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

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①

Governing law — red

A=P\left(1+\dfrac r n\right)^{nt}

$t \in \left[0, 50\right]$

$t = 0$ $A = 1000$

Along the curve

②

Domain of the variable — blue

t_{\min} = 0,\quad t_{\max} = 50

$t \in \left[0, 50\right]$

$t = 0$ $t = 0$

Along the curve

③

Response curve — green

A=P\left(1+\dfrac r n\right)^{nt}

$t \in \left[0, 50\right]$

$t = 0$ $A = 1000$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}A}{\mathrm{d}t}

$t \in \left[0, 50\right]$

$t = 0.314$ $A^{\prime} = 50.69$

Along the curve

Lecture brief

Historical brief

From al-Khwārizmī’s restoration to the quadratic formula, AM–GM, De Moivre and logarithms, algebra is the closed-form skeleton under every lab. These sheets solve, bound and compound. This sheet (ALG-14 — Compound interest) is the form associated with Compound interest. Working symbols: $P$ , $r$ , $n$ , $t$ $\rightarrow$ $A$ . n → ∞ recovers A = P e^{rt}. Effective annual rate is (1+r/n)^n − 1.

Purpose

Purpose: compute

A

from

P

r

n

t

in Algebra via

A=P\left(1+\dfrac r n\right)^{nt}

A = P (1 + r/n)^{nt}. Principal P after t years. Use it when a real algebra question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

P = 1000.000\,\mathrm{—}

r = 0.050\,\mathrm{1/yr}

n = 12.000\,\mathrm{1/yr}

t = 10.000\,\mathrm{yr}

, the governing relation

A=P\left(1+\dfrac r n\right)^{nt}

yields

A = 1647.01\,\mathrm{—}

. r as a decimal (0.05 = 5%), n compounds per year. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Principal P1000 —Nominal rate r0.05 1/yrCompounds per year n12 1/yrYears t10 yr

Outputs

Amount A1647.01 —

Reading speed

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Narration of this film

r as a decimal (0.05 = 5%), n compounds per year.

n → ∞ recovers A = P e^{rt}. Effective annual rate is (1+r/n)^n − 1.

Reading speed

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