Question

A virus attacks a single user's computer and within one hour embeds itself in 50 email attachment files sent out to other users. By the end of the hour, \(10 \%\) of these have been opened and have infected their host machines. If this process continues, how many machines will be infected at the end of 5 hours? Can you find a formula for the number of machines infected after \(n\) hours?

Short answer

At the end of 5 hours, 3906 machines will be infected. Formula: Total = 1 + \(\sum_{k=1}^{n} 5^k\) for \(n\) hours.

Step-by-step solution

Calculate machines infected each hour

The virus starts by infecting 5 new machines in the first hour, since 10% of 50 emails are opened. In formula form, this can be expressed as: \(5 = 50 \times 0.1\).

Recognize pattern of infection spread

Each newly infected machine then follows the same pattern, i.e., each machine sends 50 emails and 10% of those trigger further infections each hour.

Find infection rate formula for each subsequent hour

After each hour, the number of newly infected machines multiplies by a factor of \(5\). This is based on the previous step's pattern. We express this infectious spread for hour \(n\) using the formula for number of new infections in that hour: \(5^n\).

Calculate cumulatively infected machines after 5 hours

Sum the number of infected machines over 5 hours using the pattern: Total = 1 (initial computer) + \((5^1 + 5^2 + 5^3 + 5^4 + 5^5)\), where the powers indicate the machines infected in each hour.

Compute the total infected machines

Calculate the sum: 1 (initial) + (5^1 + 5^2 + 5^3 + 5^4 + 5^5) gives the total number of infected machines by the end of 5 hours.

Generalize infection formula for any hour n

Sum of infections until hour n: Total = 1 (initial) + \(\sum_{k=1}^{n} 5^k\). This formula represents the accumulating count of machines infected at the end of \(n\) hours.

Key concepts

Exponential Growth

Exponential growth is a process in which a quantity increases at a consistent rate over time.
In terms of a viral infection, like the one described in the exercise, the number of infected machines multiplies rapidly as each infected machine leads to more infections.
Key characteristics of exponential growth include:

• A constant percentage of current infected machines results in new infections over time.
• This creates a feedback loop, where new infections contribute to future growth.
• The rate of growth is observed in powers, such as the sequence seen in our example (e.g., 5, 25, 125).

The formula for the number of infections at hour \(n\) shows exponential characteristics: \(5^n\). This reveals the exponential nature of spread, as each hour increases the number of infections exponentially by a factor of 5.
Exponential growth models are crucial in fields such as biology, business, and technology, where rapid growth rates can significantly impact planning and management.

Mathematical Modeling

Mathematical modeling involves the creation of mathematical representations of real-world scenarios.
Such models help us predict future events and analyze complex systems.
In this exercise, modeling the spread of a virus among computers is done through a series of calculations:

• First, we identify the infection rate, which is initially 5 machines per hour based on the percentage of emails opened.
• Each hour, this model repeats, projecting further spread as per the pattern observed.
• The model sums the infected machines over time to get a cumulative total.

By deriving a general formula, like the \(\sum_{k=1}^{n} 5^k\), we can apply this model to predict the number of infections after any number of hours \(n\).
Mathematical modeling converts a complex problem into a more manageable form, enabling clear predictions and strategic decisions based on data-driven insights.

Discrete Mathematics

Discrete mathematics branches into topics that are crucial in understanding phenomena like the viral spread in our exercise.
It deals with objects that can be counted, such as the number of computers infected in a virus spread.
This field encompasses several methods used in the exercise:

• Sequences and series, seen in summing the infections over time (\(\sum_{k=1}^{n} 5^k\)).
• Geometric progression, as the infections increase by a constant factor each hour.
• Combinatorics, in managing how the infections spread with each new machine.

Discrete mathematics is vital in computing and algorithm design, as it helps solve problems that involve counting and logical structuring.
In situations like these, understanding related discrete mathematical principles provides a clear path to interpreting data, making predictions, and solving algorithmic challenges.