Problem 11 - Calculus: Early Transcendentals

Question

List the first five terms of the sequence. \( a_1 = 2, a_{n+1} = \frac {a_n}{1 + a_n} \)

Step-by-Step Solution

Answer

The first five terms are 2, \(\frac{2}{3}\), \(\frac{2}{5}\), \(\frac{2}{7}\), \(\frac{2}{9}\).

Step 1: Identify the First Term

The first term of the sequence is given as \( a_1 = 2 \). This is the initial value from which we will derive subsequent terms.

Step 2: Calculate the Second Term Using the Recursive Formula

To find the second term \( a_2 \), apply the given formula: \( a_{n+1} = \frac{a_n}{1 + a_n} \). Substitute \( a_1 = 2 \) into the formula:\[ a_2 = \frac{2}{1 + 2} = \frac{2}{3}. \]

Step 3: Calculate the Third Term

Use the formula for \( a_3 \) by substituting \( a_2 = \frac{2}{3} \):\[ a_3 = \frac{\frac{2}{3}}{1 + \frac{2}{3}} = \frac{\frac{2}{3}}{\frac{5}{3}} = \frac{2}{5}. \]

Step 4: Calculate the Fourth Term

Find \( a_4 \) by substituting \( a_3 = \frac{2}{5} \) into the formula:\[ a_4 = \frac{\frac{2}{5}}{1 + \frac{2}{5}} = \frac{\frac{2}{5}}{\frac{7}{5}} = \frac{2}{7}. \]

Step 5: Calculate the Fifth Term

Calculate \( a_5 \) by substituting \( a_4 = \frac{2}{7} \) into the formula:\[ a_5 = \frac{\frac{2}{7}}{1 + \frac{2}{7}} = \frac{\frac{2}{7}}{\frac{9}{7}} = \frac{2}{9}. \]

Step 6: List the First Five Terms

Now that all terms are calculated, list the first five terms of the sequence: \( 2, \frac{2}{3}, \frac{2}{5}, \frac{2}{7}, \frac{2}{9} \).

Key Concepts

Sequence Terms Calculation

Calculating terms in a sequence can seem challenging at first, but once you grasp the basic steps, it becomes much simpler. In sequences where each term depends on the previous one, like in the given exercise, it's crucial to start from the known initial term and apply the recipe, or formula, to find the following terms.

• **Start from the Known**: Begin with the given first term. In our exercise, this was \( a_1 = 2 \).
• **Follow the Formula**: Use the recursive relationship to find the next terms. This is your guide to deriving subsequent terms from the previous ones.
• **Step-by-Step Calculation**: Each step involves taking the last calculated term and using it to compute the next. For example, using \( a_1 = 2 \) to find \( a_2 \) as \( \frac{2}{3} \).

The power of following these steps is evident as you continue the calculations to determine each term in the sequence until the desired number is reached. Keep your calculations neat and double-check your arithmetic to avoid errors.

Recursive Formula

A recursive formula is a way to express each term of a sequence based on its previous term(s). This method of defining sequences is powerful because it transforms complex problems into smaller, manageable steps.

• **Connection Between Terms**: Unlike explicit formulas, recursive formulas relate each term directly to the one before it. This means you need to know the previous term to compute the next one.
  
• **Example in Action**: In our sequence, each term was calculated using the formula \( a_{n+1} = \frac{a_n}{1 + a_n} \). To find \( a_{n+1} \), we first calculated \( a_n \).
• **Practical Benefit**: Recursive formulas help break down calculations into smaller steps while discovering patterns and relationships between the terms.

While this approach can be computationally intensive without software, it offers a deep intuitive understanding of how sequences evolve and their inherent patterns. Recursive formulas require patience and often a systematic approach to untangle complex sequences.

Mathematical Sequences

Mathematical sequences are ordered lists of numbers and are an essential concept in mathematics, existing in various types and complexities. They help us describe patterns, relationships, and change over time in an organized manner.

• **Types of Sequences**: Sequences can be arithmetic, geometric, or recursive, among other types. Each type has its own set of rules for generating terms.
• **Applications**: Sequences appear in numerous real-world scenarios from predicting business profits, analyzing biological data, to understanding rhythms in nature.
• **Understanding the Pattern**: Recognizing the pattern of sequences is key to solving sequence problems, often simplifying complex problems into smaller, easier pieces.

Mastering how to recognize and work with different kinds of sequences equips you with the ability to handle a variety of mathematical challenges across diverse fields.