Problem 10 - Calculus: Early Transcendentals

Question

List the first five terms of the sequence. \( a_1 = 6, a_{n+1} = \frac {a_n}{n} \)

Step-by-Step Solution

Answer

The first five terms are 6, 6, 3, 1, and 0.25.

Step 1: Identify the Initial Term

The sequence starts with the initial term given as \( a_1 = 6 \). This is the first term of the sequence.

Step 2: Calculate the Second Term

Use the recurrence relation to find the second term: \( a_2 = \frac{a_1}{1} = \frac{6}{1} = 6 \).

Step 3: Calculate the Third Term

Use the recurrence relation again: \( a_3 = \frac{a_2}{2} = \frac{6}{2} = 3 \).

Step 4: Calculate the Fourth Term

Continue with the same process: \( a_4 = \frac{a_3}{3} = \frac{3}{3} = 1 \).

Step 5: Calculate the Fifth Term

Finally, calculate the fifth term: \( a_5 = \frac{a_4}{4} = \frac{1}{4} \).

Key Concepts

Recurrence Relation

In mathematics, a recurrence relation is a formula that expresses each term of a sequence in relation to one or more previous terms. This is a key concept in defining sequences where each subsequent term is calculated based on a preceding one. For example, in the sequence given by the problem, the recurrence relation is defined as:

• \( a_{n+1} = \frac{a_n}{n} \)

This relation means that to find the next term \( a_{n+1} \), you divide the current term \( a_n \) by \( n \), the position of the current term in the sequence. Recurrence relations are incredibly useful for sequences that cannot easily be described by an explicit formula. They provide a systematic way to compute each term using prior results, creating a chain of calculations that defines the entire sequence.

Initial Term

The initial term of a sequence is the first term from which the rest of the sequence is derived. It serves as the starting point for applying the recurrence relation to generate further terms. In the provided exercise:

• The initial term is given as \( a_1 = 6 \).

This term is crucial because it triggers the development of the sequence. Having the initial term defined allows the recurrence relation to be applied iteratively to find subsequent terms. Without the initial term, the sequence remains undefined because subsequent values rely on splitting from this starting point. The clarity and accuracy of defining the initial term directly impact the accuracy of the whole sequence.

Calculating Sequence Terms

Calculating sequence terms involves applying the recurrence relation methodically, starting from the initial term. For our particular sequence, this process is carried out step by step:

• The first term \( a_1 \) is 6.
• Use the recurrence relation to compute the second term: \( a_2 = \frac{a_1}{1} = 6 \).
• Continue to the third term: \( a_3 = \frac{a_2}{2} = 3 \).
• Proceed to the fourth term: \( a_4 = \frac{a_3}{3} = 1 \).
• Finally, find the fifth term: \( a_5 = \frac{a_4}{4} = 0.25 \).

Each step uses the calculation from the previous term to find the next, demonstrating the power of recurrence relations in sequence construction. This step-by-step process ensures accuracy and helps students appreciate the logical flow from one term to the next.

Mathematics Education

In mathematics education, understanding sequences and series is fundamental. Teaching students about sequences, especially through practical exercises like this one, enhances their analytical and problem-solving skills. It provides:

• Insight into how a sequence can systematically unfold from a defined starting point.
• The ability to recognize patterns and apply logical reasoning to advance through the sequence.
• Experience in applying systematic methods to solve mathematical problems.
• Skills in using recurrence relations, a central tool in discrete mathematics and algorithms.

Through exercises that challenge students to find sequence terms using recurrence relations and understand the role of initial terms, learners become equipped with valuable skills that extend beyond mathematics to all areas of logical problem-solving. These concepts knit into the bedrock of mathematical proficiency, fostering a deeper comprehension and appreciation of mathematics as a discipline.