Problem 28 - A Graphical Approach to Precalculus with Limits

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 the First Term

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

Step 2: Calculate the Second Term

The second term \(a_2\) is calculated by substituting \(n = 2\) into the recursive formula: \(a_2 = 2 \times a_{1} = 2 \times 2 = 4\).

Step 3: Calculate the Third Term

The third term \(a_3\) is found by plugging \(n = 3\) into the formula: \(a_3 = 3 \times a_{2} = 3 \times 4 = 12\).

Step 4: Calculate the Fourth Term

Using the recursive formula again for \(n = 4\): \(a_4 = 4 \times a_{3} = 4 \times 12 = 48\).

Key Concepts

Sequence Terms

In this particular sequence, the terms represent the values that are calculated at each step using the provided formula. Sequence terms are essential components in understanding the behavior of the entire mathematical sequence. We start with a given first term, in this example, the first term is \(a_1 = 2\). From there, each subsequent term is determined using a specified rule or formula.

When we mention the "first four terms" of a sequence, we're referring to the values from the first term to the fourth term in succession. For this sequence, we calculated the first term (\(a_1 = 2\)), the second term (\(a_2 = 4\)), the third term (\(a_3 = 12\)), and the fourth term (\(a_4 = 48\)).

The way these terms manifest helps us grasp the nature or pattern of the sequence, providing insight into how the terms grow or change as we move from one to the next. This concept is crucial in both arithmetic and geometric sequences, determining the progression of values.

Recursive Formula

A recursive formula is a powerful tool used to define sequences. It allows us to establish the rules for calculating any term based on previous terms. In essence, a recursive formula consists of two elements:

• The initial term (or terms) which serve as the starting point (in this case, \(a_1 = 2\)).
• A rule that relates each subsequent term to its predecessors (here, \(a_n = n \cdot a_{n-1}\) for \(n > 1\)).

In this exercise, to find a new term in the sequence, you multiply the term's position \(n\) by the preceding term \(a_{n-1}\).

This recursive nature allows for seemingly complex sequences to be broken down into simpler calculations, relying on previously computed results to build step by step. Recursive formulas are widely used in various branches of mathematics and computer science because they simplify the process of defining and calculating sequences.

Mathematical Sequences

Mathematical sequences are ordered lists of numbers which follow a set pattern or rule. They can be finite, with a limited number of terms, or infinite, continuing indefinitely. In this exercise, we're dealing with a recursive sequence, where each term is based heavily upon its predecessor.

Understanding mathematical sequences primarily involves:

• Recognizing the initial term or terms.
• Understanding how each subsequent term is generated either through a common difference (in arithmetic sequences) or a common ratio (in geometric sequences), or through other methods such as recursion.

The key benefit of sequences is their ability to model real-world phenomena, such as population growth or financial investments. Recursive sequences, like the one in this problem, highlight how a simple multiplication approach can drive exponential growth, aiding significantly in studies involving patterns and predictions. Through these structured steps, the Study of sequences offers insights into complex systems and helps foster logical reasoning.