Problem 27 - Algebra and Trigonometry with Analytic Geometry

Question

\(a_{1}=2, \quad a_{k+1}=\left(a_{k}\right)^{k}\) $$

Step-by-Step Solution

Answer

The first few terms of the sequence are 2, 2, 4, 64.

Step 1: Identify Initial Term

We are given the initial term of a sequence: \(a_1 = 2\). This is the first element of the sequence.

Step 2: Understand the Recursive Formula

The recursive formula provided is \(a_{k+1} = (a_k)^k\). This means each subsequent term is obtained by raising the previous term to the power of its position in the sequence.

Step 3: Calculate the Second Term

To find \(a_2\), apply the recursive formula with \(k = 1\): \(a_2 = (a_1)^1 = 2^1 = 2\). So, \(a_2 = 2\).

Step 4: Calculate the Third Term

To find \(a_3\), use \(k = 2\): \(a_3 = (a_2)^2 = 2^2 = 4\). Thus, \(a_3 = 4\).

Step 5: Calculate the Fourth Term

For \(k = 3\), we calculate \(a_4 = (a_3)^3 = 4^3 = 64\). So, \(a_4 = 64\).

Key Concepts

Understanding the Initial Term

When dealing with recursive sequences, the initial term is crucial. It is the starting point of a sequence from which all other terms are generated. In the exercise, the initial term is given as \(a_1 = 2\). Without this, the process of generating a sequence using a recursive formula would not commence.

The initial term acts as the foundation. It is like the seed that starts off a plant. Even though the recursive formula defines how each term relates to the previous one, the initial term is what allows the sequence to get off the ground. Without knowing \(a_1\), we couldn't determine any subsequent terms in the sequence.

Exploring the Recursive Formula

The recursive formula is a key component in generating a recursive sequence. In the given problem, it is defined as \(a_{k+1} = (a_k)^k\). This formula indicates how to generate the next term based on the current term.

Let's dissect this expression:

• \(a_{k+1}\) represents the term that follows the current term \(a_k\).
• The current term \(a_k\) is raised to the power of its position \(k\) in the sequence, highlighting the dependency on both the value and its sequence position.

This results in a powerful method to build out a sequence, where each term is intricately linked and increasingly complex as the sequence progresses. Understanding and applying the recursive formula is central to successfully calculate subsequent terms.

Carrying Out Sequence Calculation

Calculating each term in a recursive sequence involves applying the recursive formula iteratively. This process begins with the known initial term and uses the recursive formula to find subsequent terms. Let's see how this works in the provided exercise:

1. **Calculate the Second Term**: Start from \(a_1=2\) and use \(k=1\) in the recursive formula: \[a_2 = (a_1)^1 = 2^1 = 2\] So, \(a_2=2\).2. **Calculate the Third Term**: Use \(k=2\) and proceed with: \[a_3 = (a_2)^2 = 2^2 = 4\] Thus, \(a_3=4\).3. **Calculate the Fourth Term**: For \(k=3\), calculate: \[a_4 = (a_3)^3 = 4^3 = 64\] Hence, \(a_4=64\).By consistently applying the recursive formula, each term unfolds from the previous, showcasing both the power and escalating complexity of such sequences. This methodical calculation shows how recursive sequences build upon each other.