Problem 92 - Calculus for Biology and Medicine

Question

The sequence \(\mid a_{n}\) \\} is recursively defined. Compute \(a_{n}\) for \(n=1,2, \ldots, 5\) $$ a_{n+1}=2 a_{n}, a_{0}=3 $$

Step-by-Step Solution

Answer

The sequence values are: \(a_1 = 6\), \(a_2 = 12\), \(a_3 = 24\), \(a_4 = 48\), \(a_5 = 96\).

Step 1: Understand the Initial Conditions

First, note the given information: the sequence \( \{a_n\} \) begins with \( a_0 = 3 \). The recursive formula is \( a_{n+1} = 2a_n \).

Step 2: Compute the first term \(a_1\)

Using the formula \( a_{n+1} = 2a_n \) with \( n = 0 \), substitute to find \( a_1 = 2a_0 = 2 \times 3 = 6 \).

Step 3: Compute the second term \(a_2\)

Apply the recursive formula with \( n = 1 \): calculate \( a_2 = 2a_1 = 2 \times 6 = 12 \).

Step 4: Compute the third term \(a_3\)

Use the recursive formula for \( n = 2 \): find \( a_3 = 2a_2 = 2 \times 12 = 24 \).

Step 5: Compute the fourth term \(a_4\)

With \( n = 3 \), apply the formula to get \( a_4 = 2a_3 = 2 \times 24 = 48 \).

Step 6: Compute the fifth term \(a_5\)

Finally, for \( n = 4 \), calculate \( a_5 = 2a_4 = 2 \times 48 = 96 \).

Key Concepts

Initial Conditions

In the world of recursive sequences, initial conditions set the stage for everything. The initial condition tells us where to begin our journey in the sequence. For the sequence \( \{a_n\} \), it's defined that our starting point, or initial condition, is \( a_0 = 3 \). This means, quite simply, that the very first term of the sequence is 3. Understanding initial conditions is crucial because they provide the necessary groundwork for us to begin applying any recursive relations. Without a defined start, the sequence would have no context or frame of reference. Think of initial conditions as the first chapter in a story—the place where everything else begins.

Sequence Iteration

Once the initial conditions are established, the next step is to iterate through the sequence. Sequence iteration involves calculating each subsequent term one by one based on a rule or formula. In our example, each new term is calculated by doubling the previous term. This rule of sequential calculation is expressed as \( a_{n+1} = 2a_n \). Let's break it down:

• Start with \( a_0 = 3 \).
• Apply the rule for \( a_1: a_1 = 2 \times 3 = 6 \).
• Continue to \( a_2: a_2 = 2 \times 6 = 12 \).
• Next, \( a_3: a_3 = 2 \times 12 = 24 \).
• Then, \( a_4: a_4 = 2 \times 24 = 48 \).
• Finally, \( a_5: a_5 = 2 \times 48 = 96 \).

Each of these computations demonstrates the process of sequence iteration, moving step-by-step through the sequence using the recursive relationship. This process is methodical and ensures predictability and accuracy for any recursively defined sequence.

Recurrence Relations

Recurrence relations are the core formulas that connect each term in a sequence to its predecessors. They define how we generate new terms from existing ones. For the sequence \( \{a_n\} \), the recurrence relation is given by \( a_{n+1} = 2a_n \). This means that each term after the first is simply twice the term before it.Understanding recurrence relations is vital because they act as the blueprint to extend the sequence beyond the initial conditions. They establish the rule by which sequences grow and evolve. To visualize:

• Start at an initial value \( a_0 \).
• Use the recurrence relation to determine \( a_1 = 2 \times a_0 \).
• Continue using the relation for each subsequent term.

By following the recurrence relation, you ensure that each term accurately reflects the defined growth pattern of the sequence. This makes complex sequences manageable and reveals beautiful patterns within seemingly simple numbers.