Exponential Growth (Part 1)

Exponential Growth (Part 1)

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

Formula

A = A_0(1 + r)^t. A: Final Value, A_0: Initial Value, r: Rate of Change (per Period), t: Number of Period

A = A0(1 + r)t

A: finla value
A0: 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

Exponential Growth (Part 1): Proof of the Formula

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: Yearly

$1,000 investment is at a rate of 6% per year, compounded yearly. Find the estimated value of the investment 5 years later. (Assume 1.06^5 = 1.338.)

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 1.065 = 1.338.
(given condition)

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

Example 2: Monthly

$1,000 investment is at a rate of 6% per year, compounded monthly. Find the estimated value of the investment 5 years later. (Assume 1.005^60 = 1.349.)

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 1.00560 = 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

A = A_0*e^rt. A: Final Value, A_0: Initial Value, r: Rate of Change (per Period), t: Number of Period

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 = A0ert.

e = 2.718...
Just think e like π (= 3.14...).

Definition of e (mathematical constant)

Proof: Continuous Compounding

Exponential Growth (Part 1): Proof of the Continuous Compounding Formula

Start from the modified formula.

Change the exponent:
tn → (n/r)⋅rt.

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

e = 2.718...

So A = A0ert.

Example 3

$1,000 investment is at a rate of 6% per year, compounded continuously. Find the estimated value of the investment 5 years later. (Assume e^0.3 = 1.350.)

Write the given values to use.

The term 'continuously' is used.

So use the continuous compounding formula.

A = 1000⋅e0.06⋅5

Use e0.30 = 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.