# Exponential Decay (Part 1)

How to solve exponential decay problems: formulas, examples, and their solutions.

## Formula

A = A0(1 + r)t

A: finla value
A0: initial value
r: rate of change (per period)
t: number of period

The difference between
the exponential growth and decay

+r → growth
-r → decay

## Proof

Period 1

(End) = A0 - A0r
= A0(1 - r)1

Period 2

(End) = A0(1 - r) - A0(1 - r)⋅r
= A0(1 - r)(1 - r)
= A0(1 - r)2

Period 3

(End) = A0(1 - r)2 - A0(1 - r)2r
= A0(1 - r)2(1 - r)
= A0(1 - r)3

See the pattern
between the 'Period' and the 'End'.

Period t

(End) = A0(1 - r)t

Inductive reasoning

## Example 1

Write the given values to use.

Write the period unit 'min'
when writing r and t.

Put the given values into the formula.

A = 80⋅(1 - 0.05)60

Use 0.9560 = 0.046.
(given condition)

Write the answer with its unit: 3.68 g.

## Formula: Continuous Decay

For continuous decay,
use the continuous exponential decay formula:

A = A0e-rt.
(Difference: rt → -rt)

Exponential growth (part 1)

Definition of e (mathematical constant)

## Proof: Continuous Decay

Start from the formula with n.

Change the exponent:
tn → (-n/r)⋅(-rt).

Then, as n → ∞,
(1 - r/n)-n/r (dark gray) becomes e.

e = 2.718...

So A = A0e-rt.

## Example 2

Write the given values to use.

The term 'continuously' is used.

So use the continuous compounding formula.

A = 80⋅e-0.05⋅60

Use e-3 = 0.05.
(given condition)

Write the answer with its unit: 4.0 g.

As you can see,