Satyagopal Mandal
Department of Mathematics
University of Kansas
Office: 624 Snow Hall  Phone: 785-864-5180
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    Topics in Mathematics (Math105)
    Chapter 11 : Population Growth and Sequences

    The growth of population over time is a subject serious human interest. Population science considers two types of growth models - continuous growth and discrete growth. In the continuous model of growth it is assumed that population is changing (growing) continuously over time - every hour, minutes, seconds and so on. We will not discuss continuous model any further here.

    In the discrete model of growth, it is assumed that the changes in the population happen after certain regular time intervals through some transition and nothing happens between transitions.

    1. The initial size of the population is P1.
    2. After the population goes through the first transition, the size of the population is P2.
    3. After the population goes through the 2nd transition, the size of the population is P3.
    4. ... ....
    5. After the population goes through the (n-1)th transition, size of the population is Pn.
    The transition may happen in various ways:
    1. Transition may represent the entire breading season.
    2. Transition may mean simply an annual enumeration.
    3. The word "population" may also be used in a broad sense.
      1. Population may mean the collection of all FORD vehicles in US and transition may represent a full quarter.
      2. Population may mean the money in your bank account and transition may represent the full tax year.

    Remark. Recall that a list of numbers is called a sequence. The population sizes P1,P2,P3, ... forms a sequence and is called the population sequence. So, there is a lot in common between the study of population growth and sequences. Mathematical models are used to make predictions regarding population growth. In reality the growth of population is often too complex for any mathematical model to fit exactly. But realistic assumptions regarding the transition rules lead to excellent models of population growth.

    11.1 The Linear Growth Model and Arithmetic Sequence

    A simple model of growth is the linear model.

    The Linear Model
    Definition. The linear model of growth assumes that after each transition a fixed amount is added to the population.
    1. We assume that P1 is the initial size of the population and after each transition d is added to the population (d can be negative). This number d is called the common difference.
    2. So, the population size after first transition is
      P2 = P1 + d
    3. the population size after second transition
      P3 = P2 + d = P1 + 2d
    4. ... ...
    5. the population size after (n-1)th transition , the size Pn of the nth generation is given by
      Pn =Pn-1 + d
      =P1 + (n-1)d
    The Arithmetic Sequence
    The population sequence as above is an arithmetic sequence.

    Definition. A sequence a1, a2,a3, , ... , an-1,an, ... is called an arithmetic sequence if the difference between two successive terms is a fixed number d.

    1. The first term is a1. To compute the any term of the sequence we add a fixed number d to the previous term. This number d is called the common difference.
    2. So, a2 = a1 + d
    3. a3 = a2 + d = a1 + 2d
    4. ... ...
    5. the nth term an = an-1 + d
      = a1 + (n-1)d
    Sum of n consecutive terms of an
    Arithmetic Sequence
    Let a1,a2, ... , an-1,an, ... be an arithmetic sequence with common difference d. Then the sum
    a1 + a2 + ... + an-1 + an = (a1 + an )n/2
    = (first term + last term)n/2.

    Problems :SOLUTIONS.
    Problems on Linear Growth
    Exercise 11.1.1. A population grows according to linear growth model Pn = Pn -1 + 17. The initial population size is P1 = 119.
    1. What is the common difference?
    2. Compute P8.
    3. Compute P11.
    4. Compute P19.
    Exercise 11.1.2. A population grows according to linear growth model. It is known that the 8th generation population is P8 = 140 and the 15th generation population is P15 = 273.
    1. What is the common difference?
    2. Compute P1.
    3. Compute P13.
    4. Compute P21.
    Exercise 11.1.3. A population grows according to linear growth model. The first few generation populations are given by 7, 20, 33, 46, ...
    1. What is the common difference?
    2. Compute P1.
    3. Compute P9.
    4. Compute P31.
    Problems on Arithmetic Sequences
    Exercise 11.1.4. Consider the arithmetic sequence 7, 10, 13, 16, ... .
    1. What is the common difference?
    2. Compute the 9th a9.
    3. Compute the 19th term a19.
    4. Compute the 33rd term a33.
    Problems on Arithmetic Sequences
    Exercise 11.1.5. Consider the arithmetic sequence 3, 9, 15, 21, ... .
    1. What is the common difference?
    2. Compute the 10th a10.
    3. Compute the 20th term a20.
    4. Compute the 30th term a30.
    Exercise 11.1.6. Consider the arithmetic sequence an = 10+ 9(n-1).
    1. What is the common difference?
    2. Compute the 6th a6.
    3. Compute the 16th term a16.
    4. Compute the 33rd term a33.
    Problems on sum of ...
    Exercise 11.1.7. Consider the arithmetic sequence an = 10+ 3(n-1).
    1. What is the common difference?
    2. Compute the sum of first 100 terms 10 +13+16+ ... + 307.
    3. Compute the sum of first 200 terms 10 +13+16+ ... + 607
    4. Compute the sum of first 300 terms 10 +13+16+ ... + 907

    11.2 The Exponential Growth Model and Geometric Sequence

    Exponential growth model is another simplistic model of growth used in population science.

    The Exponential growth Model
    Definition. The exponential model of growth assumes that after each transition the population gets multiplied by a fixed (positive) number.
    1. We assume that P1 is the initial size of the population and after each transition the population gets multiplied by a fixed (positive) number r . This number r is called the common ratio.
    2. So, the population size after first transition is
      P2 = rP1
    3. the population size after second transition
      P3 = rP2 = r2P1
    4. ... ...
    5. the population size after (n-1)th transition , the size Pn of the nth generation is given by
      Pn =rPn-1 =rn-1P1
    The Geometric Sequences
    The population sequence as in the exponential model is a geometric sequence.

    Definition. A sequence a1, a2,a3, , ... , an-1,an, ... is called a geometric sequence if the ratio of two successive terms is a fixed number r.

    1. The first term is a1. To compute the any term of the sequence we multiply the previous term by a fixed number. This number d is called the common ratio.
    2. So, a2 = ra1
    3. a3 = ra2 = r2a1
    4. ... ...
    5. the nth term an = ran-1 = rn-1a1
    6. So, if the first term is a1 = a and the common ratio is r, the geometric sequence looks like:
      a, ra, r2a, ... ..., rn-1a, rna, ...
    Sum of n consecutive terms of an
    Arithmetic Sequence
    Let a, ra, r2a, ... ..., rn-1a, rna, ... be an geometric sequence with common ratio r and first term a. Then the sum of the first n terms is
    a + ra + r2a + ... + rn-1a = a(rn -1)/(r-1) .

    Problems: SOLUTION
    Problems on Exponential Growth
    Exercise 11.2.1. A population grows according to exponential growth model Pn = 1.7Pn -1 . The initial population size is P1 = 13.
    1. What is the common ratio?
    2. Compute P7.
    3. Compute P11.
    4. Compute P18.
    Exercise 11.2.2. A population grows according to exponential growth model. It is known that the 8th generation population is P5 = 2856100 and the 15th generation population is P14 = 30287510.66.
    1. What is the common ratio ?
    2. Compute P1.
    3. Compute P12.
    4. Compute P22.
    Exercise 11.2.3. A population grows according to geometric growth model. The first few generation populations are given by 7, 10.5, 15.75, 23.625, ...
    1. What is the common ratio ?
    2. Compute P2.
    3. Compute P8.
    4. Compute P11.
    Problems on Geometric Sequences
    Exercise 11.2.4. Consider the geometric sequence 7, 6.3, 5.67, 5.103, 4.5927, ... .
    1. What is the common ratio?
    2. Compute the 7th a7.
    3. Compute the 19th term a19.
    4. Compute the 23rd term a23.
    Exercise 11.2.5. Consider the geometric sequence 3, 3.3, 3.63, 3.993, ... .
    1. What is the common ratio?
    2. Compute the 13th a13.
    3. Compute the 23rd term a23.
    4. Compute the 33rd term a33.
    Exercise 11.2.6. Consider the geometric sequence an = 1.11(n-1)1000.
    1. What is the common ratio ?
    2. Compute the 5th a5.
    3. Compute the 15th term a15.
    4. Compute the 33rd term a33.
    Problems on sum of ...
    Exercise 11.1.7. Consider the geometric sequence an = 3.3(n-1)10.
    1. What is the common ratio?
    2. Compute the sum of first 10 terms 10 + 33 + 108.9 + ... +(3.3)910.
    3. Compute the sum of first 20 terms10 + 33 + 108.9 + ... +(3.3)1910.
    4. Compute the sum of first 30 terms10 + 33 + 108.9 + ... +(3.3)2910.
    Exercise 11.1.8. Consider the geometric sequence
    1, 0.5, 0.52,0.53,0.54,0.55, ... .
    1. What is the common ratio?
    2. Compute the sum of first 10 terms.
    3. Compute the sum of first 100 terms.
    4. Compute the sum of first 200 terms.
    5. What can you say about the sum, approximately, of a large number of terms?