Special Sequences and Mathematical Patterns

The graph curves upward because the same percentage is applied to an increasingly large population.

Interpreting the model

The model assumes:

• The percentage growth rate remains constant.

• The time periods are equal.

• There are no unexpected major changes.

• Resources and living conditions do not limit growth.

In reality, population growth may be affected by births, deaths, migration, disease, competition, and environmental conditions.

A recursive model is useful, but it is still a simplified representation of reality.

Recursive Models of Population Decline

Recursive models can also describe decreases.

Suppose a wildlife population decreases by 8% each year.

The population retains:

100% − 8% = 92%

of its previous value.

Therefore:

Pₙ₊₁ = 0.92Pₙ

Worked Example 5: Population Decline

A species has an initial population of 5,000.

The population decreases by 8% annually.

Estimate the population after three years.

P₀ = 5,000

P₁ = 5,000(0.92) = 4,600

P₂ = 4,600(0.92) = 4,232

P₃ = 4,232(0.92) = 3,893.44

Answer: Approximately 3,893 individuals.

The explicit model is:

Pₙ = 5,000(0.92)ⁿ

This is an example of exponential decay.

Population Models with a Fixed Change and Percentage Growth

Some populations change through more than one process.

For example, a town may grow naturally by a percentage each year while also receiving a fixed number of new residents.

Suppose:

• Initial population = 1,000

• Annual natural growth = 5%

• Additional residents arriving each year = 100

The recursive rule is:

Pₙ₊₁ = 1.05Pₙ + 100

This combines multiplication and addition.

Worked Example 6: Growth and Migration

Find the population after three years.

P₀ = 1,000

P₁ = 1.05(1,000) + 100

= 1,150

P₂ = 1.05(1,150) + 100

= 1,307.5

P₃ = 1.05(1,307.5) + 100

= 1,472.875

Answer: Approximately 1,473 people.

This model is more complex than a simple geometric sequence.

The percentage growth is applied first, followed by the addition of 100 residents.

Changing the order of operations may change the model.

Recursive Models in Finance

Financial processes often involve repeated percentage changes, deposits, withdrawals, and payments.

Recursive models are particularly useful because each new balance depends on the previous balance.

Compound interest

Suppose a bank account earns interest at a fixed rate every year.

Let:

Aₙ = balance after n years

A₀ = initial balance

r = annual interest rate

The recursive rule is:

Aₙ₊₁ = Aₙ(1 + r)

This is the same mathematical structure as exponential population growth.

Worked Example 7: Compound Interest

An investment begins with $2,000 and earns 5% interest annually.

Find the balance after four years.

A₀ = 2,000

Aₙ₊₁ = 1.05Aₙ

A₁ = 2,000 × 1.05 = 2,100

A₂ = 2,100 × 1.05 = 2,205

A₃ = 2,205 × 1.05 = 2,315.25

A₄ = 2,315.25 × 1.05 = 2,431.0125

Answer: $2,431.01

The explicit formula is:

Aₙ = 2,000(1.05)ⁿ

Savings Models with Regular Deposits

Many people save money by making regular deposits into an account that earns interest.

Suppose a person:

• Begins with $1,000.

• Earns 4% interest annually.

• Deposits $200 at the end of every year.

The recursive model is:

Aₙ₊₁ = 1.04Aₙ + 200

This means:

1. Calculate interest on the existing balance.

2. Add the annual deposit.

3. Use the result as the balance for the next year.

Worked Example 8: Savings with Deposits

Find the balance after three years.

A₀ = 1,000

Year 1:

A₁ = 1.04(1,000) + 200

= 1,240

Year 2:

A₂ = 1.04(1,240) + 200

= 1,489.60

Year 3:

A₃ = 1.04(1,489.60) + 200

= 1,749.184

Answer: $1,749.18

Interpreting the model

The account increases for two reasons:

• Interest is earned on the previous balance.

• A new deposit is added.

The balance does not form a simple geometric sequence because the fixed deposit changes the ratio between consecutive balances.

An explicit formula

For a constant interest rate and a fixed deposit made at the end of each period:

Aₙ = A₀(1 + r)ⁿ + d[(1 + r)ⁿ − 1]/r

where d is the deposit per period and r ≠ 0.

For the example:

A₃ = 1,000(1.04)³ + 200[(1.04)³ − 1]/0.04

= 1,124.864 + 624.32

= 1,749.184

This matches the recursive calculation.

If r = 0, the model simplifies to Aₙ = A₀ + nd.

Recursive Models for Loans

A loan balance changes through interest charges and repayments.

Suppose a borrower owes $10,000.

The loan charges 1% interest per month, and the borrower pays $500 at the end of each month.

Let Bₙ represent the outstanding balance after n payments.

The recursive rule is:

Bₙ₊₁ = 1.01Bₙ − 500

Worked Example 9: Loan Repayment

Find the outstanding balance after three payments.

Initial balance:

B₀ = 10,000

Month 1:

B₁ = 1.01(10,000) − 500

= 10,100 − 500

= 9,600

Month 2:

B₂ = 1.01(9,600) − 500

= 9,696 − 500

= 9,196

Month 3:

B₃ = 1.01(9,196) − 500

= 9,287.96 − 500

= 8,787.96

Answer: $8,787.96 remains after three payments.

Why the order matters

Interest is charged before the monthly payment is subtracted.

If the payment occurred at the beginning of the month, the recurrence would be different:

Bₙ₊₁ = 1.01(Bₙ − 500)

Real financial models must reflect the actual timing of deposits, payments, and interest.

Comparing Recursive and Explicit Models

Recursive and explicit models are two ways of describing mathematical sequences.

Recursive approach

A recursive rule calculates a term from one or more previous terms.

Example:

a₀ = 3

aₙ = 2aₙ₋₁

To find a₄:

a₁ = 6

a₂ = 12

a₃ = 24

a₄ = 48

Explicit approach

An explicit rule calculates a term directly from its position.

For the same sequence:

aₙ = 3(2)ⁿ

Therefore:

a₄ = 3(2)⁴

a₄ = 48

Comparison

When should each approach be used?

Use a recursive model when:

• The process repeats in stages.

• The next value depends on the previous value.

• Deposits, withdrawals, or other changes occur regularly.

• Different rules apply at different times.

• A simple explicit formula is unavailable.

Use an explicit model when:

• You need a distant term quickly.

• The sequence follows a known formula.

• You want to compare values at different times.

• You want to solve for the time at which a threshold is reached.

Converting Recursive Models to Explicit Models

For arithmetic and geometric sequences, conversion is straightforward.

Arithmetic sequences

Recursive:

aₙ = aₙ₋₁ + d

Initial condition:

a₀ = A

Explicit:

aₙ = A + nd

Geometric sequences

Recursive:

aₙ = raₙ₋₁

Initial condition:

a₀ = A

Explicit:

aₙ = Arⁿ

Worked Example 10: Converting an Arithmetic Model

Given:

a₀ = 12

aₙ = aₙ₋₁ − 3

The common difference is −3.

Therefore:

aₙ = 12 − 3n

Find a₁₀:

a₁₀ = 12 − 3(10)

a₁₀ = −18

Worked Example 11: Converting a Geometric Model

Given:

a₀ = 80

aₙ = 0.75aₙ₋₁

The common ratio is 0.75.

Therefore:

aₙ = 80(0.75)ⁿ

Find a₄:

a₄ = 80(0.75)⁴

= 80(0.31640625)

a₄ = 25.3125

More Advanced Recursive Models

Not all recursive models are arithmetic or geometric.

Some contain multiple operations or depend on more than one previous term.

Linear first-order recurrence

A common model is:

aₙ₊₁ = raₙ + d

This combines multiplication by a constant with addition of another constant.

It can describe savings, loans, population changes, and other repeated processes.

When r ≠ 1, its explicit form is:

aₙ = rⁿa₀ + d(rⁿ − 1)/(r − 1)

When r = 1:

aₙ = a₀ + nd

Worked Example 12: A Combined Recursive Model

A sequence is defined by:

a₀ = 10

aₙ₊₁ = 2aₙ + 3

Find a₃.

a₁ = 2(10) + 3 = 23

a₂ = 2(23) + 3 = 49

a₃ = 2(49) + 3 = 101

Answer: 101

Using the explicit formula:

aₙ = 2ⁿ(10) + 3(2ⁿ − 1)

At n = 3:

a₃ = 8(10) + 3(7)

= 80 + 21

= 101

Both approaches agree.

Recursive Models and Long-Term Behaviour

Recursive models can help predict what happens after many repeated steps.

Consider:

aₙ₊₁ = 0.8aₙ + 10

with:

a₀ = 0

The first few terms are:

a₁ = 10

a₂ = 18

a₃ = 24.4

a₄ = 29.52

a₅ = 33.616

The values increase, but the increases become smaller.

The sequence approaches a limiting value.

Finding an equilibrium value

At equilibrium, the value no longer changes.

Let the equilibrium value be L.

Then:

L = 0.8L + 10

Subtract 0.8L:

0.2L = 10

Divide by 0.2:

L = 50

The sequence approaches 50.

Important observation

A fixed point is not always approached.

For a recurrence of the form aₙ₊₁ = raₙ + d, the fixed point is generally stable when:

|r| < 1

When |r| is greater than 1, values generally move away from the fixed point unless they begin exactly there.

This connects recursive sequences with the advanced mathematical study of dynamical systems.

Recursive Models and Technology

Recursive models are also important in computer science.

Computers can generate sequences by repeatedly applying a rule.

For example, a spreadsheet can calculate the next value from the previous row.

A simple algorithm for the model:

a₀ = 5

aₙ₊₁ = 1.2aₙ

could be written as:

value = 5

repeat 5 times:
    value = value * 1.2

display value

The result is:

5(1.2)⁵ = 12.4416

This shows how recursion and iteration can be implemented computationally.

In mathematics, recursion describes a relationship using previous values.

In programming, iteration means repeatedly performing instructions. An iterative program can calculate a recursively defined sequence without a function calling itself.

Real-World Applications

Recursive models are used in many fields.

Population studies: Predicting populations through repeated births, deaths, and migration.

Finance: Calculating investment growth, savings balances, loan repayments, and depreciation.

Biology: Modelling population changes, reproduction, and repeated biological processes.

Environmental science: Modelling the amount of a pollutant remaining after repeated removal.

Medicine: Modelling drug concentrations when a fraction is eliminated between doses.

Computer science: Generating sequences and implementing repeated calculations.

Economics: Modelling changes in prices, production, and other quantities over time.

The same mathematical structure can describe very different situations.

For example:

Pₙ₊₁ = 1.05Pₙ

could represent a population growing by 5% per year.

Aₙ₊₁ = 1.05Aₙ

could represent an investment earning 5% interest per year.

The mathematics is identical, but the variables and interpretations differ.

Common Mistakes

Mistake 1: Forgetting the initial condition

A recursive rule usually requires one or more starting values.

Mistake 2: Confusing aₙ with aₙ₋₁

aₙ is the current term, while aₙ₋₁ is the preceding term.

Mistake 3: Applying a percentage incorrectly

A 6% increase means multiplying by 1.06, not by 0.06.

Mistake 4: Treating percentage growth as constant addition

A fixed percentage produces a changing absolute increase.

Mistake 5: Using the wrong number of time periods

If a₀ is the initial value, a₅ represents five completed steps.

Mistake 6: Changing the order of financial operations

Interest before a deposit is different from interest after a deposit.

Mistake 7: Assuming every recursive model is geometric

Models such as aₙ₊₁ = 1.05aₙ + 100 are not simple geometric sequences.

Mistake 8: Rounding intermediate values too early

Premature rounding can accumulate errors over many steps.

Mistake 9: Assuming recursive models always have simple explicit formulas

Some recursive models cannot be expressed using elementary explicit formulas.

Mistake 10: Extending a model without evaluating its assumptions

A constant population growth rate or interest rate may not remain realistic indefinitely.

Key Terms

• Recursive Model: Mathematical model in which new values depend on earlier values.

• Recursion: Process of defining or calculating values using previous values.

• Recursive Rule: Equation that determines a term from earlier terms.

• Initial Condition: Starting value needed to generate a recursive sequence.

• Sequence: Ordered list of numbers following a rule.

• Term: Individual value in a sequence.

• Term Number: Position or index of a term.

• Arithmetic Sequence: Sequence with a constant difference.

• Geometric Sequence: Sequence with a constant ratio.

• Common Difference: Constant amount added between arithmetic terms.

• Common Ratio: Constant multiplier between geometric terms.

• Explicit Formula: Formula that calculates a term directly from its index.

• Growth Factor: Multiplier representing a percentage increase.

• Decay Factor: Multiplier representing a percentage decrease.

• Compound Interest: Interest calculated on the principal and accumulated interest.

• Population Model: Mathematical representation of changes in population.

• Fixed Point: Value that remains unchanged when a recursive rule is applied.

• Equilibrium: State in which the model's value remains constant.

• Long-Term Behaviour: Pattern shown by a model after many repeated steps.

• Iteration: Repeated application of a calculation or procedure.

• Dynamical System: Mathematical system describing how quantities change over time.

Key Takeaways

• Recursive models describe processes in which each new value depends on earlier values.

• A recursive model requires initial conditions.

• Recursive rules can generate arithmetic, geometric, and more complex sequences.

• Arithmetic recursive models add a constant amount each step.

• Geometric recursive models multiply by a constant factor each step.

• Population growth and decay can be described recursively.

• Compound interest is a common example of a geometric recursive model.

• Savings and loans often involve both percentage changes and fixed payments.

• Recursive models are useful when changes occur in repeated stages.

• Explicit models calculate terms directly from their position.

• Recursive and explicit approaches can describe the same mathematical process.

• Some recursive models approach equilibrium values.

• Long-term predictions depend on the assumptions of the model.

• Recursive thinking is important in mathematics, science, finance, and computing.

Check Your Understanding

Questions

Question 1: Understanding Recursion

Explain what a recursive model is and why an initial condition is important.

Question 2: Generating a Sequence

Given:

a₀ = 4

aₙ = aₙ₋₁ + 7

Find a₁ through a₅.

Question 3: Geometric Recursion

Given:

a₀ = 3

aₙ = 2aₙ₋₁

Find a₄.

Question 4: Identifying the Rule

Write a recursive rule for:

8, 13, 18, 23, 28, ...

Use a₀ = 8.

Question 5: Explicit Form

Convert:

a₀ = 6

aₙ = aₙ₋₁ + 4

into an explicit formula.

Question 6: Population Growth

A town has 1,500 residents and grows by 10% each year.

Write a recursive rule and estimate the population after three years.

Question 7: Population Decline

A population of 8,000 decreases by 5% annually.

Find the population after two years.

Question 8: Compound Interest

An investment of $3,000 earns 6% annual interest.

Calculate the balance after three years.

Question 9: Savings Model

An account begins with $500.

It earns 3% interest annually, and $100 is deposited at the end of each year.

Write the recursive model and calculate the balance after two years.

Question 10: Loan Model

A loan begins at $5,000.

Interest is charged at 2% per month, followed by a payment of $300.

Find the balance after two payments.

Question 11: Comparing Models

Explain one advantage of a recursive model and one advantage of an explicit model.

Question 12: Combined Recurrence

Given:

a₀ = 2

aₙ₊₁ = 3aₙ + 1

Find a₃.

Question 13: Equilibrium

A sequence follows:

aₙ₊₁ = 0.5aₙ + 20

Find its equilibrium value.

Question 14: Modelling a Real Situation

A wildlife population begins with 600 animals.

Each year the population grows by 8%, after which 20 animals are removed.

Write a recursive rule and calculate the population after two years.

Question 15: Integrated Challenge

A savings account begins with $2,000.

The account earns 5% annual interest, and $300 is deposited at the end of each year.

a) Write the recursive model.

b) Calculate the balance after three years.

c) Write an explicit formula.

d) Use the explicit formula to verify the result.

e) Explain one assumption of the model.

Fully Worked Answers

Answer 1: Understanding Recursion

A recursive model calculates new values using one or more previous values.

An initial condition provides the starting value.

Without the initial condition, the recursive rule may describe many different sequences.

Answer 2: Generating a Sequence

Given:

a₀ = 4

aₙ = aₙ₋₁ + 7

Calculate:

a₁ = 4 + 7 = 11

a₂ = 11 + 7 = 18

a₃ = 18 + 7 = 25

a₄ = 25 + 7 = 32

a₅ = 32 + 7 = 39

Answer: 11, 18, 25, 32, 39

Answer 3: Geometric Recursion

Given:

a₀ = 3

aₙ = 2aₙ₋₁

Calculate:

a₁ = 6

a₂ = 12

a₃ = 24

a₄ = 48

Answer: a₄ = 48

Answer 4: Identifying the Rule

Sequence:

8, 13, 18, 23, 28, ...

Each term increases by 5.

Initial condition:

a₀ = 8

Recursive rule:

aₙ = aₙ₋₁ + 5, for n ≥ 1

Answer 5: Explicit Form

Given:

a₀ = 6

Common difference:

d = 4

Explicit formula:

aₙ = a₀ + nd

aₙ = 6 + 4n

Answer 6: Population Growth

Initial population:

P₀ = 1,500

Growth rate:

10% = 0.10

Recursive rule:

Pₙ₊₁ = 1.10Pₙ

Calculate:

P₁ = 1,500(1.10) = 1,650

P₂ = 1,650(1.10) = 1,815

P₃ = 1,815(1.10) = 1,996.5

Answer: Approximately 1,997 residents after three years.

Answer 7: Population Decline

Initial population:

P₀ = 8,000

Decay factor:

1 − 0.05 = 0.95

Recursive model:

Pₙ₊₁ = 0.95Pₙ

Calculate:

P₁ = 8,000(0.95) = 7,600

P₂ = 7,600(0.95) = 7,220

Answer: 7,220 individuals

Answer 8: Compound Interest

Initial investment:

A₀ = 3,000

Growth factor:

1.06

Recursive model:

Aₙ₊₁ = 1.06Aₙ

Calculate:

A₁ = 3,180

A₂ = 3,370.80

A₃ = 3,573.048

Answer: $3,573.05

Answer 9: Savings Model

Initial balance:

A₀ = 500

Interest rate:

3%

Annual deposit:

$100

Recursive rule:

Aₙ₊₁ = 1.03Aₙ + 100

Calculate:

A₁ = 1.03(500) + 100

= 615

A₂ = 1.03(615) + 100

= 733.45

Answer: $733.45

Answer 10: Loan Model

Initial balance:

B₀ = 5,000

Monthly interest:

2%

Monthly payment:

$300

Recursive rule:

Bₙ₊₁ = 1.02Bₙ − 300

First payment:

B₁ = 1.02(5,000) − 300

= 4,800

Second payment:

B₂ = 1.02(4,800) − 300

= 4,896 − 300

Answer: $4,596

Answer 11: Comparing Models

A recursive model is useful because it describes how a quantity changes from one step to the next.

An explicit model is useful because it allows a distant term to be calculated directly.

For example, an explicit formula can calculate the balance after 20 years without calculating all 19 preceding balances.

Answer 12: Combined Recurrence

Given:

a₀ = 2

aₙ₊₁ = 3aₙ + 1

Calculate:

a₁ = 3(2) + 1 = 7

a₂ = 3(7) + 1 = 22

a₃ = 3(22) + 1 = 67

Answer: a₃ = 67

Answer 13: Equilibrium

Given:

aₙ₊₁ = 0.5aₙ + 20

At equilibrium:

L = 0.5L + 20

Subtract 0.5L:

0.5L = 20

Divide by 0.5:

L = 40

The equilibrium value is 40.

Because the multiplier is 0.5, values approach this equilibrium.

Answer 14: Modelling a Real Situation

Initial population:

P₀ = 600

Growth rate:

8%

Animals removed each year:

20

Recursive model:

Pₙ₊₁ = 1.08Pₙ − 20

Calculate:

P₁ = 1.08(600) − 20

= 648 − 20

= 628

P₂ = 1.08(628) − 20

= 678.24 − 20

= 658.24

Answer: Approximately 658 animals after two years.

The model treats population as a continuous numerical quantity for calculation purposes, so rounding is appropriate when interpreting the result.

Answer 15: Integrated Challenge

Part a: Write the recursive model

Initial balance:

A₀ = 2,000

Interest rate:

5%

Annual deposit:

$300

Recursive rule:

Aₙ₊₁ = 1.05Aₙ + 300

Part b: Calculate the balance after three years

Year 1:

A₁ = 1.05(2,000) + 300

= 2,400

Year 2:

A₂ = 1.05(2,400) + 300

= 2,820

Year 3:

A₃ = 1.05(2,820) + 300

= 3,261

Balance after three years: $3,261.00

Part c: Write the explicit formula

For regular end-of-year deposits:

Aₙ = A₀(1 + r)ⁿ + d[(1 + r)ⁿ − 1]/r

Substitute:

Aₙ = 2,000(1.05)ⁿ + 300[(1.05)ⁿ − 1]/0.05

Part d: Verify the result

For n = 3:

A₃ = 2,000(1.05)³ + 300[(1.05)³ − 1]/0.05

= 2,000(1.157625) + 300(0.157625)/0.05

= 2,315.25 + 945.75

A₃ = $3,261.00

The explicit formula confirms the recursive result.

Part e: Explain an assumption

The model assumes that the interest rate remains fixed at 5% annually and that the $300 deposit is made at the end of every year.

Changes in interest rates or deposit timing would require a different model.

Final Reflection

Recursive models provide a powerful way to describe repeated processes.

They help us understand how a quantity changes from one stage to the next, whether we are investigating population growth, financial investments, savings accounts, loan repayments, or mathematical sequences.

Recursive models also connect naturally with arithmetic and geometric sequences. In many cases, an explicit formula can be derived to calculate distant terms more efficiently.

More advanced recursive models can describe systems involving several simultaneous changes or values that approach equilibrium.

The most important lesson is that recursion allows us to model change as a repeated process. By understanding how one step leads to the next, we can predict future values, analyse patterns, and make informed decisions about real-world situations.