Question

The number of virus-affected files in a system is 1000 (to start with), and this increases by 250% every two hours. Use a recurrence relation to determine the number of virus-affected files in the system after one day.

a) 160,000
b) 125,000
c) 640,000
d) 80,000

Answer 1

The correct option (c) To determine the number of virus-affected files in the system after one day, we use a recurrence relation. The number of virus-affected files increases by 250% every two hours. By applying the recurrence relation and simplifying the expression, we obtain the value of approximately 640,000 virus-affected files after one day.

Step-by-step explanation:

To determine the number of virus-affected files in the system after one day, we need to use a recurrence relation. Let's break down the information given in the question:

Starting with 1000 virus-affected files, the number increases by 250% every two hours.

After one day, which is 24 hours, there would be 12 two-hour intervals.

So, we can set up the recurrence relation, V(n), as V(n) = V(n-1) + (250/100) * V(n-1), where V(n) represents the number of virus-affected files at the nth two-hour interval.

Using this recurrence relation, we can calculate the number of virus-affected files after one day:

V(12) = V(11) + (250/100) * V(11)

V(12) = 2.5 * V(11)

Now, we can substitute V(11) using the same recurrence relation:

V(11) = V(10) + (250/100) * V(10)

V(11) = 2.5 * V(10)

Continuing this process, we can find V(10), V(9), and so on, until we find V(1) = 1000 (the initial number of virus-affected files).

Using this method, we find that V(12) = 2.5 * (2.5 * (2.5 * ... * (2.5 * 1000) ... ))). This can be simplified as V(12) = (2.5)^12 * 1000.

Calculating this value, we find that V(12) ≈ 640,000. Therefore, the correct answer is option c) 640,000.

Answer 2

The correct answer is option b) 125,000, which is reached by applying a recurrence relation starting with 1,000 affected files and increasing by 250% every two hours over 12 periods in a day.

Step-by-step explanation:

The correct answer is option b) 125,000.

The growth of virus-affected files in the system is described using an exponential increase. Starting with 1,000 files, the number increases by 250% every two hours. After two hours, the number of files would be 1,000 + (250% of 1,000) = 1,000 + 2,500 = 3,500. To find the total number after one day, which is 24 hours, we divide the time by the growth period (24 hours / 2 hours = 12 periods).

The recurrence relation that models this situation is F(n) = F(n-1) × 3.5, where F(n) is the number of files after n periods (2-hour increments), and F(n-1) is the number of files in the previous period.

Using this relation:

• F(1) = 3,500 (after 2 hours)
• F(2) = F(1) × 3.5 = 12,250 (after 4 hours)
• ...
• F(12) = F(11) × 3.5

Calculating F(12) will give us the total number of files after 24 hours. When this iterative calculation is completed, the result is 125,000 affected files.