# Exponential Growth (Part 1)

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

## Formula

*A* = *A*_{0}(1 + *r*)^{t}*A*: finla value*A*_{0}: initial value*r*: rate of change (per period)*t*: number of period

This formula is used

if a value is changing at a rate *r* for *t* periods.

## Proof

Period 1

(End) = *A*_{0} + *A*_{0}⋅*r*

= *A*_{0}(1 + *r*)^{1}

Period 2

(End) = *A*_{0}(1 + *r*) + *A*_{0}(1 + *r*)⋅*r*

= *A*_{0}(1 + *r*)(1 + *r*)

= *A*_{0}(1 + *r*)^{2}

Period 3

(End) = *A*_{0}(1 + *r*)^{2} + *A*_{0}(1 + *r*)^{2}⋅*r*

= *A*_{0}(1 + *r*)^{2}(1 + *r*)

= *A*_{0}(1 + *r*)^{3}

See the pattern

between the 'Period' and the 'End'.

Period *t*

(End) = *A*_{0}(1 + *r*)^{t}

Inductive reasoning

## Example 1: Yearly

Write the given values to use.

Write the period unit 'year'

when writing *r* and *t*.

Put the given values into the formula.*A* = 1000⋅(1 + 0.06)^{5}

Use 1.06^{5} = 1.338.

(given condition)

Write the answer with its unit: $1,338.

## Example 2: Monthly

Write the given values to use.

The unit of *r* and *t* are 'year'.

(*r* = +0.06 /year, *t* = 5 years)

But the investment is conpounded 'monthly'.

So write *n* = 12 months/year.

(12 months = 1 year)

The investment is conpounded 'monthly'.

But the unit of *r*, *t* are not 'month'.

So use the modified formula:*A* = 1000⋅(1 + 0.06/12)^{5⋅12}.

+0.06/12:

Each month,

the investment is changed

at a rate of +0.06/12.

5⋅12:

This change happens 5⋅12 times.

(5⋅12 months)

Use 1.005^{60} = 1.349.

(given condition)

Write the answer with its unit: $1,349.

Compare the answer to the previous answer.

(compounded yearly)

As *n* increases, *A* increases.

## Formula: Continuous Compounding

As *n* increases, *A* increases.

Then it is reasonable to think that

if *n* → ∞,

then *A* will be maximized.

In this case,

use the continuous compounding formula:*A* = *A*_{0}*e*^{rt}.*e* = 2.718...

Just think *e* like *π* (= 3.14...).

Definition of *e* (mathematical constant)

## Proof: Continuous Compounding

Start from the modified formula.

Change the exponent:*t*⋅*n* → (*n*/*r*)⋅*r*⋅*t*.

Then, as *n* → ∞,

(1 + *r*/*n*)^{n/r} (dark gray) becomes *e*.*e* = 2.718...

So *A* = *A*_{0}*e*^{rt}.

## Example 3

Write the given values to use.

The term 'continuously' is used.

So use the continuous compounding formula.*A* = 1000⋅*e*^{0.06⋅5}

Use *e*^{0.30} = 1.350.

(given condition)

Write the answer with its unit: $1,350.

As you can see,

the answer is greater than

the previous examples' answers:

the investment is maximized

by **continuous** compounding.