Problem 51 - Calculus for Biology and Medicine

Question

$$ \text { In Problems } \text { , write } N_{t} \text { as a function of } t \text { for each recursion } $$ $$ N_{t+1}=4 N_{t} \text { with } N_{0}=24 $$

Step-by-Step Solution

Answer

The function is \( N_t = 24 \cdot 4^t \).

Step 1: Understanding the Problem

We are given a recursive formula \( N_{t+1} = 4N_t \) with an initial value \( N_0 = 24 \). We need to express \( N_t \) as a function of \( t \), without recursion.

Step 2: Rewriting the Recursive Formula

The recursive equation \( N_{t+1} = 4N_t \) suggests that each term \( N_{t+1} \) is obtained by multiplying the previous term \( N_t \) by 4. This indicates exponential growth.

Step 3: Finding the General Term

To find a non-recursive formula, notice that each term is obtained by multiplying the initial term by 4 raised to the power of the term index. This gives the general formula: \( N_t = N_0 \cdot 4^t \).

Step 4: Substituting Initial Conditions

Substitute the initial condition \( N_0 = 24 \) into the general formula. This gives us \( N_t = 24 \cdot 4^t \).

Step 5: Conclusion

The function \( N_t \) expressed in terms of \( t \) is \( N_t = 24 \cdot 4^t \). This formula allows us to calculate \( N_t \) for any time \( t \) without recursion.

Key Concepts

Exponential Growth

Exponential growth occurs when the size of a quantity increases at a consistent rate over equal increments of time. This pattern leads to rapid growth as time progresses, as each step involves multiplying the previous value by a fixed factor. In our example, we observe increasing growth by a factor of 4. This is because the recursive formula given is \( N_{t+1} = 4N_t \). The multiplier here is 4, which signifies an exponential growth pattern.

• Small initial amounts can grow to large numbers very quickly due to the multiplicative nature.
• In the context of the given problem, starting from 24 (our initial condition), and multiplying by 4 successively, results in the size increasing exponentially.

Exponential growth can frequently be seen in various natural and financial processes, indicating its vast applicability.

Non-Recursive Solutions

A non-recursive solution, unlike recursive ones, does not depend on prior terms to compute the next term. It provides a direct mathematical expression for the nth term of a sequence, allowing for ease of calculation without referencing prior terms.

• In our context, we derived a non-recursive formula: \( N_t = 24 \cdot 4^t \).
• This formula succinctly calculates \( N_t \) directly for any given \( t \) without going through each previous step.

Utilizing non-recursive solutions can greatly simplify the process of computation, making it efficient to determine values at higher steps.

Initial Conditions in Sequences

Initial conditions set the starting point of a sequence and greatly influence its development. For sequences following a recursive formula, such as \( N_{t+1} = 4N_t \), the initial condition determines the scale of all subsequent values in the sequence.

• In our problem, the initial condition provided is \( N_0 = 24 \).
• This specific value is crucial as it acts as the base from which all further computations and sequence values are derived.
• Substituting this initial condition into our non-recursive formula \( N_t = N_0 \cdot 4^t \) results in \( N_t = 24 \cdot 4^t \), showcasing the influence of initial conditions on the sequence.

Initial conditions are key in tailoring general sequences to specific scenarios or problems.