Problem 10 - Essential Calculus

Question

Write the first five terms of the recursively defined sequence. \(a_{1}=6, a_{k+1}=\frac{1}{3} a_{k}^{2}\)

Step-by-Step Solution

Answer

The first five terms of the sequence are: 6, 12, 48, 768, 196,608.

Step 1: Identify the First Term

We can see that the first term is already given in the problem as \(a_{1}=6\). This will be the starting point for the sequence.

Step 2: Compute the Second Term

The second term can be computed using the formula \(a_{k+1}=\frac{1}{3} a_{k}^{2}\). Plugging in \(k=1\) into the formula and using \(a_{1}=6\), we get \(a_{2}=\frac{1}{3} a_{1}^{2} = \frac{1}{3} * 6^2 = 12\).

Step 3: Compute the Third Term

Similar to step 2, plug in \(k=2\) into the formula and use \(a_{2}=12\) to get \(a_{3}=\frac{1}{3} a_{2}^{2} = \frac{1}{3} * 12^2 = 48\).

Step 4: Compute the Fourth Term

Repeat the process described in step 2 and 3. Plug in \(k=3\) into the formula and use \(a_{3}=48\). This yields \(a_{4}=\frac{1}{3} a_{3}^{2} = \frac{1}{3} * 48^2 = 768\).

Step 5: Compute the Fifth Term

Similar to before, plug in \(k=4\) into the formula and use \(a_{4}=768\). This will get us the fifth term \(a_{5}=\frac{1}{3} a_{4}^{2} = \frac{1}{3} * 768^2 = 196,608\).

Key Concepts

Sequence Computation

When tasked with calculating terms of a sequence, it involves a straightforward but repetitive method of applying the given rule to the known values. In this exercise, to find the values of the sequence, you start from the initial term. Which is provided as the first number in the sequence.
The process includes:

• Identifying the initial term, which serves as our base for computing subsequent terms.
• Using the sequence rule to compute each subsequent term based on the preceding one.

This particular sequence is defined so that each term depends on the one before it, relying on mathematical operations that make each step unique, yet structured.
For this sequence, each term is calculated using the relationship given as \( a_{k+1} = \frac{1}{3} a_{k}^2 \). Understanding how to apply these operations correctly forms the core of sequence computation.

Mathematical Recursion

Recursion is a method of solving problems where the solution involves solving smaller sub-problems of the same type. In sequences, mathematical recursion happens when each term of a sequence is defined in terms of the previous term through a function or equation. It's like solving a puzzle where each piece helps to reveal the next one.

• The initial term (known or given) kick-starts the sequence.
• Each subsequent term is defined using the recursive formula.

With the sequence we've explored, the recursive formula is \(a_{k+1} = \frac{1}{3} a_{k}^{2}\). This means you can directly compute each term once you know the previous one.
Recursion requires careful application of the formula, ensuring that the transformation from one term to the next correctly reflects the rule you've been given. This helps create a series of terms without listing each one explicitly from the start.

Sequence Terms

A sequence consists of numbers in a particular order, dictated by a specific rule or formula. Each number in the sequence is called a sequence term, and these individual terms are calculated step by step using the criteria laid out by the recursive definition.

• The first term is usually given and acts as a reference point.
• Subsequent terms are generated by applying the recursive formula repeatedly.

Each term connects mathematically to its predecessor through operations like addition, subtraction, multiplication, or in complex series, factorial or powers as seen here.
By starting with the known term, recalculating through each step ensures that steps are not skipped, and prevents errors in generating the sequence. As a result, this guides you to maintain accuracy and understand how sequences expand over iterations.