Question

The population of City A is 8,000,000 at the end of the year 2020.The

number of immigrants is 25,000n at the end of year n. The population

of city increases at the rate of 8% per year. Use recurrence relation to

determine the population of the city at the end of 2030.

Answer 1

Solution

The population of the city at the end of the year 2020 = 8,000,000

Let us set the variable “n” (the time in years) as

At the end of year 2020, n = 0

At the end of the year 2021, n = 1

At the end of the year 2022, n = 2

....

At the end of the year 2030, n = 10

Since, at the end of the year 2020, the population is 8,000,000

 

At the end of the year 2021 the population will be

P = 8,000,000 + 1× 8% of 8,000,000 + 25,000 × 1

P = 8,000,000 × 1.08 + 25,000

P (1) = 8,000,000 × (1.08)¹ + 25,000(1)

 

At the end of the year 2022

P = 8,000,000 × (1.08)¹ × (1.08)¹ + 25, 000 × 2

P (2) = 8,000,000 × (1.08)² + 25, 000 × 2

 

Similarly, at the end of the year 2023

P = 8,000,000 × (1.08)¹ × (1.08)¹ × (1.08)¹ + 25, 000 × 3

P (3) = 8,000,000 × (1.08)³ + 25, 000 × 3

…

We can generalize the formula as

$P\left( n \right) = 8,000,000 \times {\left( {1.08} \right)^n} + 25,000 \times n$

Hence,

And at the end of the year 2030

$P\left( {10} \right) = 8,000,000 \times {\left( {1.08} \right)^{10}} + 25,000 \times 10 \\$

$P\left( {10} \right) = 17521399.98\\$

$P\left( {10} \right) \approx 17521400 \\$