Problem 91 - Calculus for Biology and Medicine

Question

In Problems 91-100, the sequence \(\mid a_{n}\) \\} is recursively defined. Compute \(a_{n}\) for \(n=1,2, \ldots, 5\) $$ a_{n+1}=2 a_{n}, a_{0}=1 $$

Step-by-Step Solution

Answer

The sequence is \( a_1 = 2 \), \( a_2 = 4 \), \( a_3 = 8 \), \( a_4 = 16 \), \( a_5 = 32 \).

Step 1: Understand the Recursive Formula

The problem gives us a recursive formula: \( a_{n+1} = 2a_n \). This means each term in the sequence is twice the previous term. The initial term of the sequence is \( a_0 = 1 \).

Step 2: Calculate \( a_1 \)

Using the recursive formula, we compute the first term:\[ a_1 = 2 imes a_0 = 2 imes 1 = 2 \]

Step 3: Calculate \( a_2 \)

Using the recursive formula, we compute the second term:\[ a_2 = 2 imes a_1 = 2 imes 2 = 4 \]

Step 4: Calculate \( a_3 \)

Using the recursive formula, we compute the third term:\[ a_3 = 2 imes a_2 = 2 imes 4 = 8 \]

Step 5: Calculate \( a_4 \)

Using the recursive formula, we compute the fourth term:\[ a_4 = 2 imes a_3 = 2 imes 8 = 16 \]

Step 6: Calculate \( a_5 \)

Using the recursive formula, we compute the fifth term:\[ a_5 = 2 imes a_4 = 2 imes 16 = 32 \]

Key Concepts

Sequence Calculation

In mathematics, sequences are essentially lists of numbers arranged in a specific order. Each number in the list is called a "term." To calculate the sequence, we typically start with an initial term and apply a rule to find the following terms. These calculations help us identify the pattern that the sequence follows.

Consider the sequence from our exercise, which is derived using a rule or formula. Given:

• Initial term: \( a_0 = 1 \)
• Recursive formula: \( a_{n+1} = 2a_n \)

We apply the recursive formula to compute each subsequent term. For example:

• For \( a_1 \): We multiply the initial term \( a_0 \) by 2, so \( a_1 = 2 \times a_0 = 2 \times 1 = 2 \).
• For \( a_2 \): We take \( a_1 \) and again multiply by 2, resulting in \( a_2 = 2 \times 2 = 4 \).
• This pattern is repeated for each term, making the sequence obvious once the first few terms are calculated.

Recursive Formula

The recursive formula is the backbone of defining and calculating sequences. It allows us to generate an entire sequence by strictly defining the step-by-step process. The formula given in the exercise illustrates this beautifully:

• \( a_{n+1} = 2a_n \)

This formula tells us how to calculate the next term (\( a_{n+1} \)) using the current term (\( a_n \)). By implementing a simple multiplication operation, the entire sequence is built from just the initial term. Recursive formulas are convenient because they don't require starting from scratch every time. They merely extend the existing calculations to find what follows in the sequence.

Let's visualize this: you start with \( a_0 \), then use the formula repeatedly to find each subsequent term. The recursive formula is particularly useful for sequences that have patterns like geometric progressions or repeated multiplication.

Initial Term

The initial term is a foundational element in any sequence. It acts as the seed from which the rest of the sequence is derived. Our given exercise presents the initial term as \( a_0 = 1 \).

Here's why the initial term matters:

• It determines the starting point of the sequence.
• All other terms are calculated based on it.
• If the initial term changes, the entire trajectory of the sequence would change.

In recursive sequences, the initial term is indispensable because it provides the first value the recursive formula acts upon. Essentially, while the formula guides how subsequent terms are calculated, the initial term is the origin. In our sequence, without \( a_0 = 1 \), determining \( a_1, a_2, a_3, a_4, \) and so on, would be impossible.