Problem 34 - Calculus Single Variable

Question

In Exercises 34-40, give a recursive definition of the sequence. $$ f_{n}=2 n, n=1,2,3, \ldots $$

Step-by-Step Solution

Answer

Initial condition: \( f_1 = 2 \). Recursive rule: \( f_n = f_{n-1} + 2 \).

Step 1: Understand the problem

We need to provide a recursive definition for the sequence \( f_n = 2n \), which means expressing each term in the sequence in terms of its predecessor.

Step 2: Identify the initial condition

The sequence starts at \( n=1 \). The initial term is \( f_1 = 2 \times 1 = 2 \). This serves as the starting point for our recursive definition.

Step 3: Recognize the pattern

The sequence is an arithmetic sequence where each term increases by 2 compared to the previous one, i.e., \( f_{n} = 2n \). We can express the next term recursively as \( f_{n} = f_{n-1} + 2 \).

Step 4: Write the recursive definition

Based on the pattern, the recursive definition is:1. Initial condition: \( f_1 = 2 \)2. Recursive step: \( f_n = f_{n-1} + 2 \) for \( n > 1 \). This captures the rule that each term is 2 more than the previous term.

Key Concepts

Arithmetic Sequence

An arithmetic sequence is a series of numbers in which the difference between any two successive members is always the same. This difference is called the "common difference." In the case of the sequence given in the exercise, this sequence is a classic example of an arithmetic sequence. When we look at the sequence defined by the formula \( f_n = 2n \), it transforms into \( 2 \), \( 4 \), \( 6 \), \( 8 \), ..., with each term consistently increasing by 2 from its predecessor. Some essential points about arithmetic sequences include:

• The formula for an arithmetic sequence can often be expressed as \( a_n = a_1 + (n-1) \, d \), where \( a_1 \) is the first term, and \( d \) is the common difference.
• Understanding the constant increase provides insight into the behavior of the sequence as it progresses.
• Recognizing arithmetic patterns helps in identifying relationships between numbers, crucial for solving mathematical problems efficiently.

Sequence Definition

The term "sequence" refers to an ordered list of numbers or elements arranged according to a specific rule or pattern. In mathematics, sequences can be finite or infinite, and providing a definition aids in understanding the structure and rules governing the sequence. The given sequence \( f_n = 2n \) is defined by a simple multiplicative rule, making it easy to predict and calculate subsequent terms. Here, each element is explicitly defined rather than inferred from the preceding one till a recursive definition translates this implicit rule into a practical format.

• Explicit Form: This gives the \( n^{th} \) term directly, as is done in \( f_n = 2n \).
• Recursive Definition: In this form, each term is defined as a function of the preceding term(s). Such definitions are particularly powerful in computer programming and algorithms.
• Understanding both forms allows for flexible ways to approach sequence problems, choosing what's most applicable to the context.

Initial Condition

The initial condition in a sequence denotes the start point that sets the stage for all following terms. It gives the base value from which others are derived based on the defined rule or formula. Identifying and understanding the initial condition is crucial for establishing a sequence correctly. In our sequence \( f_n = 2n \), the initial condition is established as \( f_1 = 2 \). This first term ensures that the recursive function \( f_n = f_{n-1} + 2 \) is correctly applied to compute all subsequent terms.

• An accurate initial condition is vital as it affects all subsequent elements in a sequence. A shift or error can lead to an entirely different sequence.
• It acts as both a starting point for computation and a reference for validation of the sequence's integrity.
• Essential in illustrating sequences as a whole, offering insight into patterns from a beginning point.