Problem 36 - Calculus for Biology and Medicine

Question

Find an expression for \(a_{n}\) on the basis of the values of \(a_{0}, a_{1}, a_{2}, \ldots\) \(0,1,2,0,1,2\)

Step-by-Step Solution

Answer

The expression is \( a_n = \begin{cases} 0 & \text{if } n \equiv 0 \mod 3 \\ 1 & \text{if } n \equiv 1 \mod 3 \\ 2 & \text{if } n \equiv 2 \mod 3 \end{cases} \).

Step 1: Identify the Pattern

Observe the given sequence: \(0, 1, 2, 0, 1, 2\). Notice that the sequence repeats every three terms. This is indicative of a periodic pattern with a period of 3.

Step 2: Define the General Term

Since the sequence repeats every three terms, we can express the general term \(a_n\) in terms of \(n\). The value of \(a_n\) depends on \(n\mod 3\) (the remainder when \(n\) is divided by 3).

Step 3: Determine Value Based on Remainder

Calculate \(a_n\) for each case of \(n\mod 3\):- If \(n \equiv 0 \mod 3\), then \(a_n = 0\).- If \(n \equiv 1 \mod 3\), then \(a_n = 1\).- If \(n \equiv 2 \mod 3\), then \(a_n = 2\).

Step 4: Construct the Expression

Based on the values determined from the modulo operation, the expression for \(a_n\) is:\[a_n = \begin{cases} 0 & \text{if } n \equiv 0 \mod 3 \1 & \text{if } n \equiv 1 \mod 3 \2 & \text{if } n \equiv 2 \mod 3\end{cases}\]

Key Concepts

Periodic Patterns

In mathematics, a sequence is often characterized by identifiable patterns that repeat at regular intervals. This repetition is what we refer to as a periodic pattern. For the sequence given in the exercise, which is \(0, 1, 2, 0, 1, 2\), we observe that it repeats every three terms, making it a periodic sequence with a period of 3. This means that after every third term, the sequence goes back to its initial terms and starts a new cycle. Understanding periodic patterns is crucial in mathematics because it allows us to predict future elements of a sequence based on these repeating cycles.

Recognizing these patterns can simplify complex problems, turning them into easier, predictable processes. It is often the first step in solving sequence-related problems, as it sets the foundation for finding expressions that describe the general term.

Modular Arithmetic

Modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after reaching a certain value known as the modulus. This concept is essential for solving sequences that demonstrate periodic behavior since it helps to determine the unique values within each cycle of the sequence.

In the given problem, the sequence repeats every three terms. Hence, the modulus here is 3. When calculating \(n \mod 3\), you're finding what the remainder is when you divide \(n\) by 3. This remainder will be 0, 1, or 2—precisely reflecting the values in each cycle of the sequence. Modular arithmetic allows for identifying which number in the repeating sequence should appear at any given position \(n\). This helps to simplify and solve sequence problems using clear mathematical logic.

Remember, using mod is like looping back to the start after hitting the maximum, much like hours on a clock.

General Term Expression

Once you've identified the periodic pattern and understood the role of modular arithmetic, constructing the general term expression becomes straightforward. The general term gives us a formula to find any term in the sequence based on its position \(n\).

For the sequence presented, we rely on the concept that different scenarios arise depending on the results of \(n \mod 3\). The general term \(a_n\) can be expressed using a conditional structure:

• If \(n \equiv 0 \mod 3\), then \(a_n = 0\).
• If \(n \equiv 1 \mod 3\), then \(a_n = 1\).
• If \(n \equiv 2 \mod 3\), then \(a_n = 2\).

With this expression, we neatly capture the behavior of the periodic sequence, showing how the value of any term in the sequence is determined simply by its position modulo 3. This approach streamlines the process of working with periodic sequences, making it efficient and clear to understand.