Problem 98 - Calculus for Biology and Medicine

Question

The sequence \(\mid a_{n}\) \\} is recursively defined. Compute \(a_{n}\) for \(n=1,2, \ldots, 5\) $$ a_{n+1}=\sqrt{a_{n}}, a_{0}=16 $$

Step-by-Step Solution

Answer

\( a_1 = 4, a_2 = 2, a_3 \approx 1.414, a_4 \approx 1.189, a_5 \approx 1.090 \).

Step 1: Identify the Initial Term

The sequence is defined recursively, starting with the initial term. We are given that \( a_{0} = 16 \).

Step 2: Calculate the First Term

Using the recursive formula \( a_{n+1} = \sqrt{a_{n}} \), calculate \( a_{1} \). \[ a_{1} = \sqrt{a_{0}} = \sqrt{16} = 4 \] So, \( a_{1} = 4 \).

Step 3: Calculate the Second Term

Now, calculate the next term \( a_{2} \) using \( a_{1} = 4 \).\[ a_{2} = \sqrt{a_{1}} = \sqrt{4} = 2 \] So, \( a_{2} = 2 \).

Step 4: Calculate the Third Term

Continue by calculating \( a_{3} \) using \( a_{2} = 2 \).\[ a_{3} = \sqrt{a_{2}} = \sqrt{2} \approx 1.414 \] So, \( a_{3} \approx 1.414 \).

Step 5: Calculate the Fourth Term

Next, compute \( a_{4} \) using \( a_{3} \approx 1.414 \).\[ a_{4} = \sqrt{a_{3}} = \sqrt{1.414} \approx 1.189 \] So, \( a_{4} \approx 1.189 \).

Step 6: Calculate the Fifth Term

Finally, determine \( a_{5} \) using \( a_{4} \approx 1.189 \).\[ a_{5} = \sqrt{a_{4}} = \sqrt{1.189} \approx 1.090 \] So, \( a_{5} \approx 1.090 \).

Key Concepts

Understanding Recursive Formula

A recursive formula is a way to define a sequence by establishing a relationship between its successive terms. In simpler terms, it means each term is determined from the previous one.
This is crucial in sequences where you calculate each term using the one before it. For instance, in our exercise, the formula given is:

• \(a_{n+1} = \sqrt{a_{n}}\)

The subscript \(n+1\) represents the next term, while \(n\) is the current term.
This shows how each new term is a transformation, specifically a square root, of the previous one.
Using a recursive formula like this can help calculate even complex sequences in an organized manner. It offers step-by-step progression from a known starting point.

Decoding Sequence Terms

Sequence terms refer to the ordered numbers in a series like \(a_0, a_1, a_2, ...\). In recursive sequences, each term signifies a step in the progression from an initial value.
In the given problem, we start with an initial term \(a_0 = 16\). The following terms are computed using the recursive formula:

• \(a_1 = \sqrt{16} = 4\)
• \(a_2 = \sqrt{4} = 2\)
• \(a_3 = \sqrt{2} \approx 1.414\)
• \(a_4 = \sqrt{1.414} \approx 1.189\)
• \(a_5 = \sqrt{1.189} \approx 1.090\)

Observing the sequence terms helps identify patterns or stability points in the sequence. Here, as you see, each term gets progressively smaller as it nears a value. This tells us about the convergence pattern of the sequence.

Grasping Initial Term Calculation

The initial term of a sequence is your starting point. It forms the foundation for all subsequent terms. For recursive sequences, knowing the initial term is crucial as the entire sequence builds off of this value.
In our exercise, the initial term is given as \(a_0 = 16\). From this, each subsequent term is derived by repeatedly applying the recursive formula:
\(a_{n+1} = \sqrt{a_{n}}\)
This initial term needed no calculation and was provided outright. However, in practice, the initial term could arise from a specific condition or formula.
Given the significance of the initial term, proper identification and use is essential to correctly proceeding with calculating the full sequence. Understanding how each term connects back to the initial sets a strong base to rely on as you dive into recursive patterns.