Problem 15 - Precalculus with Limits

Question

In Exercises 9-32, write the first five terms of the sequence. (Assume that \( n \) begins with 1.) \( a_n = \dfrac{n + 2}{n} \)

Step-by-Step Solution

Answer

The first five terms of the sequence \( a_n = \dfrac{n+2}{n} \) are 3, 2, 5/3, 1.5, and 1.4.

Step 1: Substituting n into the sequence formula

Firstly, let's substitute \( n = 1 \) in the formula \( a_n = \dfrac{n+2}{n} \) to find the first term. Therefore, \( a_1 = \dfrac{1+2}{1} = 3 \)

Step 2: Find the second term

Next, substitute \( n = 2 \) in the formula \( a_n = \dfrac{n+2}{n} \). So, \( a_2 = \dfrac{2+2}{2} = 2 \)

Step 3: Find Third Term

Then, substitute \( n = 3 \) in the formula \( a_n = \dfrac{n+2}{n} \). So, \( a_3 = \dfrac{3+2}{3} = \frac{5}{3} \)

Step 4: Find Fourth Term

Now, substitute \( n = 4 \) in the formula \( a_n = \dfrac{n+2}{n} \). So, \( a_4 = \dfrac{4+2}{4} = 1.5 \)

Step 5: Calculate the Fifth Term

Finally, substitute \( n = 5 \) in the formula \( a_n = \dfrac{n+2}{n} \). So, \( a_5 = \dfrac{5+2}{5} = 1.4 \)

Key Concepts

Finding Terms of a Sequence

Sequences are an essential concept in precalculus, and finding the terms of a sequence involves understanding and applying a rule that defines the relationship between sequential elements. For instance, consider the sequence defined by the formula
\( a_n = \dfrac{n + 2}{n} \).
To find the terms of this sequence, you have to substitute consecutive natural numbers for \(n\), which represents the position of each term in the sequence.

By plugging in the first few natural numbers (starting from 1), you find the corresponding terms of the sequence:
\(a_1 = 3\), \(a_2 = 2\), \(a_3 = \frac{5}{3}\), \(a_4 = 1.5\), and \(a_5 = 1.4\).
This step-by-step approach is fundamental when the sequence is not immediately recognizable, or it doesn't follow a common pattern like arithmetic or geometric progressions.

Arithmetic Sequences

Arithmetic sequences, one of the most straightforward types of sequences, are defined by a constant difference between consecutive terms. This difference is known as the 'common difference,' denoted as \(d\). For an arithmetic sequence, the nth term is computed using the formula
\(a_n = a_1 + (n-1)d\),
where \(a_1\) is the first term.

However, the sequence in our original exercise, \(a_n = \dfrac{n + 2}{n}\), is not an example of an arithmetic sequence because the difference between terms changes from one pair of consecutive terms to another. Recognizing whether a sequence is arithmetic is crucial for applying the correct methods and formulas to find additional terms or solve problems related to such sequences.

Fractional Sequences

Fractional sequences are sequences where each term is a fraction that can involve variables or constants. The sequence given by \(a_n = \dfrac{n + 2}{n}\) is a perfect illustration of a fractional sequence.

Unlike arithmetic sequences, fractional sequences may not follow a linear pattern but can still exhibit a regular behavior that allows for the prediction of future terms. To analyze these sequences:

• Check whether the numerator and denominator have terms that increase or decrease with \(n\).
• Observe how changes in \(n\) affect the overall value of the fraction.
• Determine if there are common patterns or simplifications that can be made.

For this particular sequence, the terms approach 1 as \(n\) increases because the numerator grows at the same rate as the denominator, causing the fraction to get closer to the value of 1.