Problem 48 - Calculus for Biology and Medicine

Question

$$ \text { In Problems } \text { , write } N_{t} \text { as a function of } t \text { for each recursion } $$ $$ N_{t+1}=2 N_{t} \text { with } N_{0}=7 $$

Step-by-Step Solution

Answer

The function is \( N_{t} = 7 \times 2^{t} \).

Step 1: Understanding the Recursion Relation

The given recursion formula is \( N_{t+1} = 2N_{t} \) with an initial condition \( N_{0} = 7 \). This formula tells us that to get the next term, we multiply the current term by 2.

Step 2: Finding the First Few Terms

Start with \( N_{0} = 7 \). Compute subsequent terms as follows: \( N_{1} = 2N_{0} = 2 \times 7 = 14 \), \( N_{2} = 2N_{1} = 2 \times 14 = 28 \), and so on. This shows a pattern where each term is a power of 2 times the initial term.

Step 3: Identifying the Pattern

From the calculations, we observe that the terms follow the pattern \( N_{t} = 7 \times 2^{t} \). This format represents the general solution as a function of \( t \).

Step 4: Writing the General Formula

The pattern leads to the general solution: \( N_{t} = 7 \times 2^{t} \). This expression encapsulates the relationship between \( N_{t} \) and \( t \) for the given recursion relation.

Key Concepts

Geometric Sequences

In mathematics, a sequence in which each term after the first is found by multiplying the previous term by a fixed nonzero number is known as a geometric sequence.
The fixed number is referred to as the common ratio. In the context of our problem, the recursion relation \( N_{t+1} = 2N_{t} \) showcases a geometric sequence where the common ratio is 2.
Each term is derived by multiplying the preceding term by this common ratio.

• This means in a geometric sequence, the terms can be expressed using a general formula: \( a_n = a_1 imes r^{(n-1)} \) where \( a_n \) is the \( n \)-th term, \( a_1 \) is the first term, and \( r \) is the common ratio.
• In our example, the first term \( N_0 \) is 7, and the common ratio \( r \) is 2. Therefore, each subsequent term can be determined as \( N_t = 7 \times 2^t \), indicating a consistent structure for the sequence.

Initial Conditions

Initial conditions are fundamental in defining sequences as they provide the starting point for recursion relations.
They are particularly vital in recursive formulas because knowing one term is necessary to find subsequent terms.
For our recursion relation \( N_{t+1} = 2N_{t} \), the initial condition given was \( N_0 = 7 \).

• This value sets the first term in the sequence and guides the direction and growth of the entire sequence.
• Without an initial condition, the specific values of the sequence could not be determined.Thus, initial conditions are operational as they inform the specific trajectory and population of the sequence.
• Once the starting point is known, each further term relies directly upon its predecessors, effectively building from the initial condition.

Exponential Growth

Exponential growth occurs when the rate of increase of a quantity is proportional to its current value, often illustrated through equations and sequences like our example.
The recursion relation we explored, \( N_{t+1} = 2N_{t} \), exhibits exponential growth, meaning each term is a fixed multiple of the one before it.

• This is represented by the general formula we derived: \( N_t = 7 \times 2^t \).
• Here, the base of the exponential \( b \) in the expression \( 2^t \) indicates that the sequence doubles each time \( t \) increases by 1.
• This kind of growth is rapid as the increase accelerates with each additional term, reflecting how populations and investments might grow exponentially under consistent conditions.

Understanding exponential growth helps comprehend broader mathematical concepts in fields such as finance, biology, and computer science.