Problem 52 - Calculus for Biology and Medicine

Question

$$ \text { In Problems } \text { , write } N_{t} \text { as a function of } t \text { for each recursion } $$ $$ N_{t+1}=5 N_{t} \text { with } N_{0}=17 $$

Step-by-Step Solution

Answer

The function is \( N_t = 5^t \times 17 \).

Step 1: Understand the Recursive Formula

The given recursive formula is \( N_{t+1} = 5N_t \), meaning that each subsequent term is 5 times the previous term. We also have an initial condition \( N_0 = 17 \). Our task is to express \( N_t \) as an explicit function of \( t \).

Step 2: Derive the First Few Terms

To see the pattern, start calculating the first few terms using the recursive relation: \( N_1 = 5 \cdot N_0 = 5 \cdot 17 = 85 \), then \( N_2 = 5 \cdot N_1 = 5 \cdot 85 = 425 \), and continue this pattern to find \( N_3 = 5 \cdot 425 = 2125 \).

Step 3: Identify the Pattern and Formulate a General Expression

From the calculations, we observe that \( N_1 = 5^1 \cdot 17 \), \( N_2 = 5^2 \cdot 17 \), and \( N_3 = 5^3 \cdot 17 \). The pattern suggests that \( N_t = 5^t \cdot 17 \).

Step 4: Express \( N_{t} \) as a Function of \( t \)

We have identified that the expression for \( N_t \) in terms of \( t \) is \( N_t = 5^t \cdot 17 \). This formula will hold for any non-negative integer \( t \).

Key Concepts

Explicit Function

An explicit function is a direct formula used to express a sequence in terms of its position, without needing to refer back to previous terms. It provides a clear and straightforward way to calculate any term in a sequence. In this exercise, the explicit function for the sequence is \( N_t = 5^t \cdot 17 \). This formula tells us exactly how to calculate the value of \( N_t \) for any term number \( t \).

The advantage of having an explicit function is that it allows quick computation of any term without calculating all the preceding terms.

For example, if you want to find \( N_{10} \), you can simply substitute \( t = 10 \) into the explicit formula:

• Compute \( 5^{10} \)
• Multiply it by 17

This approach is efficient, especially for large values of \( t \), and removes the need for repetitive calculations.

Recursive Formula

A recursive formula defines each term in a sequence based on its preceding terms. In the case of this problem, the recursive formula is \( N_{t+1} = 5N_t \). This means that each term is obtained by multiplying the previous term by 5.

Recursive formulas can be very intuitive as they closely follow the process of generating the sequence naturally, but they require you to know the value of the previous term.

In this sequence, starting with the known term \( N_0 = 17 \), you can continually apply the recursive relation to find subsequent terms:

• \( N_1 = 5 \times 17 = 85 \)
• \( N_2 = 5 \times 85 = 425 \)
• \( N_3 = 5 \times 425 = 2125 \)

This step-by-step process illustrates how each next term is derived from the previous, demonstrating the power and flow of recurrence in sequences.

Initial Condition

The initial condition in a recursive sequence is the starting point or the first known term of the sequence. It is crucial for calculating subsequent terms when using a recursive formula. For this exercise, the initial condition is \( N_0 = 17 \).

Without this initial condition, you would not have a starting point from which to apply the recursive formula. The initial condition serves as the foundation upon which the entire sequence is built.

Typically, initial conditions provide:

• The base value needed to begin applying the recursive rule.
• The specific starting point for calculating the sequence.
• A way to align the sequence uniquely according to its problem context.

In this example, knowing \( N_0 \) allows us to calculate all future terms through either the recursive or explicit formula approach. Starting sequences without initial conditions is like trying to solve a puzzle without all the pieces.