Satyagopal Mandal
Department of Mathematics
University of Kansas
Office: 624 Snow Hall  Phone: 785-864-5180
  • e-mail: mandal@math.ukans.edu
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    Exponential Growth Model:

     

    A population growth model is called a Exponential growth model if after each transition the population size of the population gets multiplied by a FIXED multiple r.

     

      1. The multiple r is called the common ratio.

        2.  

      2. So, we have

     

    P1 = Starting population

     

    P2 = rP1

     

    P3 = rP3 = r2P1

    ……….

     

    PN = rPN-1 = r N-1P1.

     

     

    Remark :

    1) If

    R > 1 then the population size PN grows very rapidly

     

    2) And if

    0 < r < 1, then the population size PN decays.

     

     

     

    Definition: Any sequence of numbers that behave like the exponential growth model as above is called a GEOMETRIC sequence. So, a geometric sequence looks like

     

    a, ar, ar2, ar3, ….., arN-1, arN, …..

     


     

    Ex.30 (Page 354). A population grows according to an exponential growth model. The starting population is P1=8 and the common ratio r=1.5.

      1. Find P2

      2. Find P10

      3. Give an explicit description for the population sequence.

     

     

     

    Adding N consecutive terms of a Geometric Sequence:

     

    a + ar +ar2+ar3+…..+arN-1= a(1-rN-1)/(1-r)

     

     

    Ex. 32. Consider the geometric sequence with first 4 terms 1,3,9,27.

      1. Find P100

      2. Find PN

      3. Find P1+P2+…+P100

      4. Find P50+P51+…+P100

     

     

    How Does Your Money Grow? …..

     

    Example A. Suppose your father invested $10,000 in an account at your birth for your college. The bank pays 12 percent annual interest. Interest is paid once a year at the end of the year. How much money you will have on your 18th Birthday? (Will you have enough money for college?)

     

    Example Q. Suppose the interest is paid at the each quarter. How much will you have on your 18th birthday?

     

    Example. M. Suppose interest is paid at the end of each month. How much money will you have on you 18th birthday?

     

    Formula : Suppose you invest $P1 in a bank account. The bank pays an annual interest as a decimal I (so 100I percent). Interest is compounded k times a year. Then the balance in your account at the end of N years is

     

    PN+1=P1rNk=P1(1+i/k) Nk.

     

    Definition. Annual (Percentage) Yield is the percentage increase the account will produce in 1 year.

    So,

    Annual Yield=100((1+i/k)k-1) percent

     

     

    Exercise 24.a) (changed) The amount of $874.00 is deposited in a account that pays 12 percent interest compounded daily. Assuming no withdrawals are made, how much money will be in the account after 2 years?

    b) What is the annual yield on this account?

     

    Here P1=874,

    i=annual interest rate =0.12

    k=number of times the interest is compounded in a year = 365

     

    So, P3 = P1(1+i/k) Nk =874(1+.12/365)2x365=1111.02

     

    Annual Yield = 100((1+i/k)k-1)=

     

    100((1+.12/365)365-1)=12.75 percent