This is a beta course, so its structure, chapters, and examples may continue to change.

Population Growth

Population growth is a classic application of sequences, from simple exponential growth models to complex Logistic models, all involving sequence knowledge.

The Exponential Growth Model

The simplest population growth model assumes: the population growth rate is fixed each year.

Let the population in year $n$ be $P_n$ and the annual growth rate be $r$ . Then:

$P_{n+1} = P_n(1+r)$

This is a geometric sequence!

Exponential growth formula

P_n = P_0(1+r)^n

where $P_0$ is the initial population.

Practical Applications

Example 1: City Population Forecast

A city had a population of 1 million in 2020 with an annual growth rate of 2%. Forecast the population in 2030.

Solution:

$P_{10} = 100 \times 1.02^{10} \approx 121.9 \text{ (ten thousand)}$

Example 2: Bacterial Reproduction

A certain bacterium divides once every hour (doubling in number). If there are initially 1000 bacteria, how many are there after 6 hours?

Solution:

$P_6 = 1000 \times 2^6 = 64000 \text{ bacteria}$

Example 3: Doubling Time

With an annual population growth rate of 3%, how many years until the population doubles?

Solution:

$P_0 \times 1.03^n = 2P_0$ $1.03^n = 2$ $n = \frac{\ln 2}{\ln 1.03} \approx 23.4 \text{ years}$

The Logistic Model (Optional)

In reality, populations do not grow without bound; they are limited by resources. The Logistic model takes the environmental carrying capacity $K$ into account:

$P_{n+1} = P_n + rP_n\left(1 - \frac{P_n}{K}\right)$

As $P_n$ approaches $K$ , the growth rate tends to 0.

Epidemic Transmission Models

The spread of epidemics such as COVID-19 can also be modeled similarly:

$I_{n+1} = I_n \cdot R_0 \cdot \left(1 - \frac{I_n}{N}\right)$

where $R_0$ is the basic reproduction number, $N$ is the total population, and $I_n$ is the number of infected people on day $n$ .

The terror of exponential growth: If

R_0 = 2

(each infected person infects 2 others), after 10 generations the number of infections is

2^{10} = 1024

times! This is why epidemic prevention must emphasize “early detection and early isolation”.

Practice Problems

Exercise 1

A country had a population of 50 million in 2020 with an annual growth rate of 1.5%. Forecast the population in 2050.

Reference Answer(2 个标签)

sequence applicationpopulation growth

Solution:

$P_{30} = 5000 \times 1.015^{30} \approx 7816 \text{ (ten thousand)}$

Answer: about 78.16 million

Exercise 2

A certain virus spreads at a daily rate of 50% (each infected person infects 0.5 people on average). Initially 10 people are infected. About how many people are infected after 7 days? (Assume the total population is large enough)

Reference Answer(2 个标签)

sequence applicationpopulation growth

Solution:

Simplified model (ignoring environmental carrying capacity):

$I_7 = 10 \times 1.5^7 \approx 171 \text{ people}$

Answer: about 171 people

Summary

Symbols Used in This Article

符号	类型	读音/说明	在本文中的含义
$P_n$	数学符号	P-sub-n	The population in year $n$
$P_0$	数学符号	P-sub-zero	The initial population
$r$	数学符号	r	The annual growth rate
$n$	数学符号	n	Number of years
$K$	数学符号	K	The environmental carrying capacity
$R_0$	数学符号	R-sub-zero	The basic reproduction number
$N$	数学符号	N	The total population
$I_n$	数学符号	I-sub-n	The number of infected people on day $n$
$\ln$	数学符号	natural log	The natural logarithm

中英对照

中文术语	英文术语	音标	说明
指数增长	exponential growth	/ˌekspəˈnenʃəl ɡrəʊθ/	Growing at a fixed proportion
增长率	growth rate	/ɡrəʊθ reɪt/	The percentage of growth per period
承载力	carrying capacity	/ˈkæriɪŋ kəˈpæsəti/	The maximum amount the environment can sustain
递推关系	recurrence relation	/rɪˈkʌrəns rɪˈleɪʃən/	A formula that derives the next term from the previous one
基本再生数	basic reproduction number	/ˈbeɪsɪk ˌriːprəˈdʌkʃən ˈnʌmbə/	The average number of people infected by a single infected person

Loan Repayment

Geometric Problems