Problem 16
Question
Write the next two apparent terms of the sequence. Describe the pattern you used to find these terms. \(\frac{7}{2}, 4, \frac{9}{2}, 5, \ldots\)
Step-by-Step Solution
Verified Answer
The next two terms in the sequence are: 5.5, 6
1Step 1: Identifying the Pattern
First, rewrite the whole numbers as fractions: \(\frac{7}{2}, \frac{8}{2}, \frac{9}{2}, \frac{10}{2}\). Now, it becomes clear that the numerator of the fraction is incrementing by 1 for each subsequent term.
2Step 2: Finding the Next Term
To find the next term, we increase the numerator of the last known term (which is 10) by 1, making it 11. So, the next term would be \(\frac{11}{2}\) or 5.5 when written in decimal form.
3Step 3: Finding the Second Next Term
Following the same method, to find the term after \(\frac{11}{2}\), we will increase the numerator by 1, making it 12. So, the subsequent term will be \(\frac{12}{2}\), which simplifies to 6.
Key Concepts
Arithmetic ProgressionFraction NotationPattern Recognition
Arithmetic Progression
Sequences like the one in the exercise are often associated with arithmetic progression. In a typical arithmetic progression, each term is derived from the previous term by adding a constant difference. The sequence might not seem immediately like an arithmetic progression due to its fraction notation. However, once rewritten using a common format, it becomes evident that the sequence aligns with this concept.
For example, consider the sequence: \(\frac{7}{2}, \frac{8}{2}, \frac{9}{2}, \frac{10}{2}\). The constant difference here is \(\frac{1}{2}\). By recognizing this, you can correctly predict upcoming terms by simply continuing to add \(\frac{1}{2}\) to the numerator.
For example, consider the sequence: \(\frac{7}{2}, \frac{8}{2}, \frac{9}{2}, \frac{10}{2}\). The constant difference here is \(\frac{1}{2}\). By recognizing this, you can correctly predict upcoming terms by simply continuing to add \(\frac{1}{2}\) to the numerator.
- This approach emphasizes the pattern of evenly increasing numerators, typical of arithmetic progressions.
- Even when numbers appear differently, reducing them to a common form can simplify the identification of such patterns.
Fraction Notation
Understanding fractions is crucial when dealing with sequences expressed in this form. Fraction notation helps represent numbers that are not whole, clearly demonstrating the relationships between parts and wholes.
In the given sequence, fractions are used to facilitate a straightforward comparison of each number's magnitude. By rewriting integers as fractions, such as 4 being represented as \(\frac{8}{2}\), analyzing the sequence becomes more accessible. This is because:
In the given sequence, fractions are used to facilitate a straightforward comparison of each number's magnitude. By rewriting integers as fractions, such as 4 being represented as \(\frac{8}{2}\), analyzing the sequence becomes more accessible. This is because:
- The numerators are emphasized, allowing a more comfortable visual understanding of the sequence's progression.
- In cases like this, simplifying fractions can also help in linking them to their whole-number counterparts, thus enhancing comprehension.
Pattern Recognition
Recognizing patterns is the cornerstone of deciphering and predicting sequences. Pattern recognition involves observing regularities or trends in data. In sequences, recognizing such patterns enables you to anticipate future terms even with incomplete information.
In our sequence, identifying that the numerator is consistently progressing by one is key to predicting upcoming terms:
In our sequence, identifying that the numerator is consistently progressing by one is key to predicting upcoming terms:
- The exercise illustrates how breaking down elements into familiar forms can reveal the underlying structure, returning us to pattern recognition.
- Upon spotting this, one can continuously forecast the sequence by applying the established pattern.
Other exercises in this chapter
Problem 15
Determine the convergence or divergence of the series. $$ \sum_{n=1}^{\infty} \frac{(-1)^{n+1} n !}{1 \cdot 3 \cdot 5 \cdot \cdot \cdot(2 n-1)} $$
View solution Problem 15
Use the power series $$\frac{1}{1+x}=\sum_{n=0}^{\infty}(-1)^{n} x^{n}$$ to determine a power series, centered at 0 , for the function. Identify the interval of
View solution Problem 16
Use the Integral Test to determine the convergence or divergence of the series, where \(k\) is a positive integer. $$ \sum_{n=1}^{\infty} n^{k} e^{-n} $$
View solution Problem 16
In Exercises \(7-28,\) find the interval of convergence of the power series. (Be sure to include a check for convergence at the endpoints of the interval.) $$ \
View solution