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# Sequences project

Module by: Pinelands High School. E-mail the author

## Grade 11 – Project: Sequences

### Introduction

The spread of disease, particularly in Africa, is a cause of extreme concern, not only for Africans but also for the rest of the world. Infectious diseases, which are passed from one human being to another, can spread at an alarming rate. The rate of infection is dependent on a number of factors and we will look at different diseases spreading at different rates.

Halting the spread of disease is best done not just by medicine but by the control of unhealthy practices such as having multiple sexual partners or drinking unpurified water. The following project seeks to bring across the importance of acting in a responsible manner and educating those who are unaware of the potentially dangerous situation they may be a part of.

1) Let us assume that a deadly virus, AD12, starts off with one host, and analyse the effects of contact that person has with others both directly and indirectly. If one person passes the virus to two people in what we will call the 2nd infection and those two pass it on to two more people in the 3rd infection and so on, we develop a pattern of numbers to represent the total number of infections:

1 + 2 + 4 + …

a) How many people will get the virus in each of the next three infections if the pattern of each person infecting two more people continues?

4th infection:

5th infection:

6th infection:

b) Determine the “ formula” for the nth term, Tn , of this sequence of numbers where n represents the stages of infection and Tn the number of people infected during the nth stage.

A second virus, IP65, also spreads but at a different rate. Research shows that from an initial sample of 1 person, the virus spreads according to the pattern below with ten people affected in the second infection, twenty five in the third and so on.

1 + 10 + 25 + 46 + …

a) How many people will get the virus IP65 in each of the next three infections if the pattern continues?

5th infection:

6th infection:

7th infection:

b) Describe the pattern in words.

c) Determine the nth term, Tn, of this sequence of numbers. Show all working.

3) A third virus, RR87, also spreads at a different rate with the following infection levels measured:

1 + 19 + 37 + 55 + ...

a) How many people will get this virus in each of the next three infections if the pattern continues?

5th infection:

6th infection:

7th infection:

b) Describe the pattern in words.

c) Determine the nth term, Tn, of this sequence of numbers.

We are now going to investigate the spread of these three viruses when compared with each other by graphing the formulas found in 1b, 2c and 3c in Task 1.

Draw, on the axes below, the graphs of the three formulas with n represented on the horizontal axis and Tn on the vertical axis. Use a scale of 1 block = 20 people on the vertical axis and graph up to the 10th infection.

From the graph:

1) Which virus is potentially more deadly in terms of its rate of spreading?

2) Give a reason why you chose the virus above.

Of far greater significance perhaps is the total number of people with the virus in each case. We will now investigate ways of finding formulas to determine those totals.

Eg: Consider a pattern of 2 + 8 + 14 + 20 + ...

If we use the symbol S to denote the sum of all the terms, then S4 = 44 in this pattern. (Check to see how this number was obtained)

1) For each of the viruses, calculate the total number of infections after each stage up to the fifth infection. Fill in the rest of the table below.

2) Plot the total number of infections at each stage from the data you have calculated above with n on the horizontal axis and Sn on the vertical axis, using the scale of 1 block = 10 people on the vertical axis.

3) Now determine a formula for calculating the total infections for the three viruses below in terms of n, the infection stages. (Show all calculations please)

IP65:

RR87:

1a) 8; 16; 32

b) Tn=2n1Tn=2n1T_n = 2^{n - 1}

2a) 73; 106; 145

b) Add 9, then 15, then 21. Increase by 6 each time.

c) Tn=3n22Tn=3n22T_n = 3n^{2} - 2

3a) 73; 91; 109

b) Add 18 to get next term

c) Tn=18n17Tn=18n17T_n = 18n - 17

2) Exponential increase

1)

2)

3)

AD12 : S n = 2 n 1 IP65 : S n = n 3 + 3 2 n 2 3 2 n RR87 : S n = 9n 2 8n AD12 : S n = 2 n 1 IP65 : S n = n 3 + 3 2 n 2 3 2 n RR87 : S n = 9n 2 8n AD12: S_n = 2^n - 1 newline IP65: S_n = n^3 + {3 over 2} n^2 - {3 over 2}n newline RR87: S_n = 9n^2 - 8n
(1)

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