Problem 53 - Calculus for Biology and Medicine

Question

$$ \text { In Problems } \text { , write } N_{t} \text { as a function of } t \text { for each recursion } $$ $$ N_{t+1}=\frac{1}{2} N_{t} \text { with } N_{0}=5000 $$

Step-by-Step Solution

Answer

The function is \(N_t = \frac{5000}{2^t}\).

Step 1: Understand the Recursion Formula

The problem gives a recursive formula: \(N_{t+1} = \frac{1}{2}N_{t}\). This formula means that each term in the sequence is half of the previous term. You also have the initial condition \(N_{0} = 5000\).

Step 2: Expand the Sequence

Using the recursion formula, calculate the first few terms.1. \(N_{1} = \frac{1}{2} \times 5000 = 2500\)2. \(N_{2} = \frac{1}{2} \times 2500 = 1250\)3. \(N_{3} = \frac{1}{2} \times 1250 = 625\)This shows a pattern where each term is half of the previous term.

Step 3: Generalize the Pattern as a Function

The pattern shows that each term is half of the previous, starting from 5000. The general formula for \(N_{t}\) can be written as:\[N_t = \frac{5000}{2^t}\]This is because for each increment in \(t\), \(N_{t}\) is divided by 2.

Step 4: Verify the Function

Verify the function by plugging in values of \(t\):- For \(t = 0\), \(N_0 = \frac{5000}{2^0} = 5000\) (matches the initial condition)- For \(t = 1\), \(N_1 = \frac{5000}{2^1} = 2500\)- For \(t = 2\), \(N_2 = \frac{5000}{2^2} = 1250\)The function holds true for these points, confirming it accurately represents the recursion.

Key Concepts

Sequence

A sequence is an ordered list of numbers that follow a particular pattern or rule. In the context of mathematics, it helps to look at sequences as a list where each element, called a term, is defined based on its position.
For example:

• The sequence can be simple like 1, 2, 3, 4... where each number increases by 1.
• Alternatively, it can be more complex, like the sequence in our problem: 5000, 2500, 1250, 625...

In this exercise, every number or term in the sequence is derived from the one before it.
This process results in a specific order, where each element follows from its predecessor according to a predetermined rule. Thus, understanding sequences is crucial since they lay the groundwork for recognizing patterns in numbers.

Initial Condition

An initial condition is the starting point of a sequence. It's like the foundation or the first step that sets the rest of the sequence in motion.
In the problem, the initial condition is given as \(N_0 = 5000\).
This initial value is important because it is the term from which all other terms in the sequence will be derived.Having an initial condition is majorly helpful because:

• It allows us to determine the sequence's direction.
• Knowing the initial condition helps in predicting subsequent terms.
• Any changes in the initial value will directly affect the entire sequence.

Thus, setting the correct initial condition ensures the accuracy of the resulting sequence.

Recursive Formula

A recursive formula is a rule or equation that shows how to find the next term in a sequence using the previous term(s). This formula builds the entire sequence step by step.
In our problem, the recursive formula is \(N_{t+1} = \frac{1}{2}N_t\). This conveys that each new term, \(N_{t+1}\), is simply half of the term before it.The recursive formula's role includes:

• Helping you understand the concept of change from one term to the next.
• Clarifying how each term depends on its predecessor, creating a systematic progression.
• Allowing us to continue the sequence as far as needed without redoing previous steps.

Understanding recursive formulas forms a bridge to more advanced mathematics, where such patterns are vital.