Problem 61 - Precalculus with Limits

Question

In Exercises 47-62, write an expression for the apparent \(n\)th term of the sequence. (Assume that \( n \) begins with 1.) \( 1 + \dfrac{1}{1}, 1 + \dfrac{1}{2}, 1 + \dfrac{1}{3}, 1 + \dfrac{1}{4}, 1 + \dfrac{1}{5}, \dots \)

Step-by-Step Solution

Answer

The apparent \(n\)th term of the sequence is \(1 + \dfrac{1}{n}\).

Step 1: Recognize the Pattern

The first step is to recognize the pattern in the sequence. Looking at the terms of the sequence, we see that each term is a sum of 1 and the reciprocal of the term number.

Step 2: Create a Formula

With the pattern identified, a formula for the nth term of the sequence can now be constructed. The formula will be \(1 + \dfrac{1}{n}\), where \(n\) is the term number.

Key Concepts

nth Term

Finding the nth term of a sequence involves identifying a single expression that represents every term in the sequence. This concept is crucial when dealing with arithmetic sequences as it provides a systematic way to describe the entire sequence with one formula.

For the sequence given:

• ext{First term:} \( 1 + \frac{1}{1} \)
• ext{Second term:} \( 1 + \frac{1}{2} \)
• ext{Third term:} \( 1 + \frac{1}{3} \)

We notice each term follows a consistent structure: it is a combination of 1 and a fraction that consists of the reciprocal of the term number. Thus, we can deduce that the nth term can be expressed as:
\[ a_n = 1 + \frac{1}{n} \]

This expression succinctly substitutes any integer value for \( n \) to find the corresponding term in the sequence. By understanding the nth term, you can now predict any future term without needing to manually extend the sequence.

Pattern Recognition

Pattern recognition is a fundamental skill in mathematics that involves observing a sequence or set of numbers to identify a consistent rule or pattern. In arithmetic sequences, this typically means finding a repeated structure or relationship among the terms.

By examining the sequence: \( 1 + \frac{1}{1}, 1 + \frac{1}{2}, 1 + \frac{1}{3}, \dots \), you quickly notice a pattern: each term increases by decreasing the denominator of a fraction that is added to 1. Recognizing this pattern allows us to identify:

• 1 is a constant part of every term.
• The denominator of the fractional part is equal to the position of the term in the sequence.

This intuitive understanding helps simplify complex sequences into manageable forms, making it easier to create formulas like the one for the nth term. Pattern recognition not only aids in solving sequences but also enhances problem-solving abilities across various mathematical topics.

Reciprocal

Understand the reciprocal as a fundamental concept in mathematics also aids in understanding the sequence presented. The reciprocal of a number is simply 1 divided by that number, and it plays a pivotal role in this particular arithmetic sequence.

For instance, in our sequence:
- the reciprocal of 1 is \( \frac{1}{1} \)- the reciprocal of 2 is \( \frac{1}{2} \)- the reciprocal of 3 is \( \frac{1}{3} \)

Reciprocals transform numbers to alternative useful representations, especially when combined with other mathematical operations. By recognizing that each term in our sequence includes the reciprocal of its position number, you can understand how these contribute to forming each term.

This concept isn't just about fractions but about how numbers relate to one another through division. It complements your ability to see and work with patterns, making reciprocal a key piece in understanding and applying mathematical concepts efficiently.