Problem 49 - Calculus for Biology and Medicine

Question

$$ \text { In Problems } \text { , write } N_{t} \text { as a function of } t \text { for each recursion } $$ $$ N_{t+1}=3 N_{t} \text { with } N_{0}=12 $$

Step-by-Step Solution

Answer

The function is \( N_t = 12 \times 3^t \).

Step 1: Understanding the Initial Condition

The problem provides us with an initial value for the sequence. Here, we know that the starting value of the sequence is given by \( N_0 = 12 \). This value will be used to calculate subsequent values of the sequence.

Step 2: Analyzing the Recurrence Relation

The recurrence relation \( N_{t+1} = 3N_t \) indicates that each term of the sequence is obtained by multiplying the previous term by 3. Knowing this allows us to build a pattern and derive a general formula.

Step 3: Identifying the Pattern

To find a general formula for \( N_t \), consider the first few terms:- \( N_0 = 12 \)- \( N_1 = 3 \times N_0 = 3 \times 12 = 36 \)- \( N_2 = 3 \times N_1 = 3 \times 36 = 108 \)We see that each term is 3 times the previous term.

Step 4: Deriving the General Formula

By recognizing the multiplication pattern, we can generalize \( N_t \) as a function of \( t \) as follows: \[ N_t = N_0 imes 3^t \]Substitute the initial condition into this formula:\[ N_t = 12 imes 3^t \]

Step 5: Conclusion

We have derived \( N_t \) as a function of \( t \): The sequence is represented by \( N_t = 12 \times 3^t \). Each term is determined by multiplying 12 by \( 3 \) raised to the power of \( t \).

Key Concepts

Initial Condition

In the world of recurrence relations, the initial condition is like the starting point of a journey. It tells you where to begin. For this particular exercise, the initial condition is given by \( N_0 = 12 \). This means that the first term in the sequence, when \( t = 0 \), is 12.

The initial condition is crucial because every subsequent term in the sequence is built upon this starting value. It serves as an anchor point, letting you calculate the sequence without ambiguity. Imagine if someone tells you they are baking a cake and they start halfway – you’d be puzzled, right? Similarly, without the initial condition, you can’t properly chart the progress of the sequence.

• Starting value: \( N_0 = 12 \)
• Sets the base for further calculations

Recognizing the initial condition enables us to effectively utilize the recurrence relation and derive a comprehensive understanding of the sequence.

General Formula

When dealing with sequences defined by recurrence relations, the ultimate goal is often to find a general formula. This formula allows us to determine any term in the sequence directly, without recursing through all previous terms.

For our problem, the recurrence relation is \( N_{t+1} = 3N_t \), which indicates that each term is obtained by multiplying the previous term by 3. With the initial condition \( N_0 = 12 \), you can derive the general formula like this:
- Start with \( N_1 = 3 \times N_0 \)
- Then \( N_2 = 3 \times N_1 = 3^2 \times N_0 \), and so on.

This pattern leads to the general formula:
\[ N_t = N_0 \times 3^t \]

This formula allows us to find \( N_t \) directly by plugging in any \( t \). It's like having a universal recipe for any term in the sequence, saving you time and computation.

• General formula: \( N_t = 12 \times 3^t \)
• Provides direct computation of terms

Using the general formula is an efficient way to work with sequences.

Multiplication Pattern

The multiplication pattern observed in recurrence relations is key to understanding how sequences operate. In our exercise, each term is a result of multiplying the previous term by 3. This unbroken chain of multiplication forms a predictable pattern.

Let's break down the pattern:
- Start with \( N_0 = 12 \)
- The next term, \( N_1 = 3 \times N_0 = 36 \)
- Then, \( N_2 = 3 \times N_1 = 108 \)

This pattern, where each term is a multiple of the prior one by a constant factor (here 3), makes predictions straightforward. Understanding this allows us to appreciate why the general formula is \( N_t = 12 \times 3^t \).

Here is why the multiplication pattern is important:

• Shows how each term is connected to its predecessor
• Enables the derivation of the general formula
• Predicts future values with a simple calculation

Mastering the multiplication pattern is a powerful tool in managing and predicting sequences governed by recurrence relations.