Problem 24 - A Graphical Approach to College Algebra

Question

Find the first four terms of each sequence. $$a_{1}=2, a_{n}=n \cdot a_{n-1}, \text { for } n>1$$

Step-by-Step Solution

Answer

The first four terms are 2, 4, 12, and 48.

Step 1: Identify First Term

The first term of the sequence is given as \(a_1 = 2\). This is the starting point for finding the sequence.

Step 2: Calculate Second Term

The second term is calculated using the formula \(a_n = n \cdot a_{n-1}\) with \(n = 2\). Thus, \(a_2 = 2 \times a_1 = 2 \times 2 = 4\).

Step 3: Calculate Third Term

To find the third term, apply the formula with \(n = 3\). Therefore, \(a_3 = 3 \times a_2 = 3 \times 4 = 12\).

Step 4: Calculate Fourth Term

For the fourth term, use \(n = 4\) in the given formula so \(a_4 = 4 \times a_3 = 4 \times 12 = 48\).

Key Concepts

Recursive Formula

A recursive formula is a way of defining terms in a sequence using the preceding terms. In the given exercise, the sequence is defined through a formula: \( a_n = n \cdot a_{n-1} \), where \( a_1 = 2 \) is the first term. This means that each term is dependent on the term immediately before it.

The approach here is different from an explicit formula, where each term is defined clearly without dependence on previous terms. Recursive formulas are particularly useful for sequences where it might be tedious or complex to write out an explicit formula.

• Identify the base case, or the first term, which is crucial as it acts as the starting point for the entire sequence.
• Apply the recursive rule using the given formula to generate subsequent terms, each calculated based on its immediate predecessor. Just as in the exercise where terms are calculated stepwise: from \( a_1 = 2 \), we calculate \( a_2 \), then from \( a_2 \), we find \( a_3 \), and so on.
• The primary advantage is its simplicity in defining complex relationships.

Sequence Terms

Every sequence is composed of different terms, often referred to by their position in the sequence like first term, second term, and so forth. In our case, the task was to calculate the first four terms of a particular sequence.

To identify and calculate these sequence terms properly, follow these steps:

• Start with the initial or first term, \( a_1 = 2 \). This acts as our foundation.
• Use the recursive relationship to find subsequent terms like \( a_2, a_3, a_4 \), each calculated based on its predecessor. Applying the formula systematically allows you to build the sequence.

For example:

• Second Term: Calculated using \( a_2 = 2 \cdot a_1 = 4 \).
• Third Term: Derived from \( a_3 = 3 \cdot a_2 = 12 \).
• Fourth Term: Found using \( a_4 = 4 \cdot a_3 = 48 \).

The sequence visually represents how each term builds on the previous, making patterns easier to see and understand.

Mathematical Induction

Mathematical induction is a powerful technique for proving statements or formulas regarding numbers or sequences. In the context of a recursive sequence, induction could be used to generalize a formula for an entire sequence or to prove properties about it.

The principle of mathematical induction involves two fundamental steps:

• Base Case: Prove that the statement holds for the initial term, usually \( n=1 \). In our exercise, this is evident since \( a_1 = 2 \) is given and already valid.
• Inductive Step: Assume the statement is true for \( n=k \), proving that it must also be true for \( n=k+1 \). This requires demonstrating that if \( a_k = k \cdot a_{k-1} \) holds, then \( a_{k+1} = (k+1) \cdot a_k \) follows logically.

When solving arithmetic sequences with inductive proof, each step confirms the formula's validity for progressing terms of the sequence. While the current exercise does not explicitly ask for induction, understanding it helps deepen comprehension of recursive patterns and their proofs.