Problem 14 - Advanced Calculus

Question

Given \(x_{n}=3 n+(-1)^{n}(n-5)+7\), (a) Calculate \(x_{1}, x_{2}, \ldots, x_{10}\). (b) Find all the numbers that ever appear twice in the entire sequence. *(c) Do any terms appear three times?

Step-by-Step Solution

Answer

a) The first 10 terms of the sequence are: 5, 10, 13, 18, 21, 26, 29, 34, 37, 42. b) No numbers ever appear twice in this sequence. c) No terms appear three times in the sequence.

Step 1: STEP 1: COMPUTE THE TERMS

In this step, you substitute 1, 2, 3, ..., 10 into the sequence \(x_{n}=3 n+(-1)^{n}(n-5)+7\) to get the first 10 terms. Calculate \(x_{n}\) for all \(n\) from 1 to 10.

Step 2: STEP 2: IDENTIFY DUPLICATE TERMS

Having calculated the first 10 terms, you now need to identify if there are any duplicate values among them. Write down all the terms and look for values that appear more than once.

Step 3: STEP 3: VERIFY TERMS APPEARING THREE TIMES

The final step involves identifying if there are any terms in the sequence that appear three times. Go through the list of terms already computed and count the frequency of each term. If any term appears three times, then it satisfies the condition.

Key Concepts

Calculating Sequence Terms

Understanding how to calculate terms in a mathematical sequence is essential for analyzing patterns and solving problems. In our given exercise, the sequence is defined by an explicit formula: \(x_{n} = 3n + (-1)^{n}(n - 5) + 7\). To calculate the terms \(x_{1}, x_{2}, ..., x_{10}\), we plug in the values of \(n\) from 1 to 10 into the formula.

For example, to find the first term \(x_{1}\), substitute \(n=1\) into the formula to get \(x_1 = 3(1) + (-1)^1(1 - 5) + 7\). Similarly, proceed with \(n=2\) through \(n=10\) and calculate each corresponding term. This systematic approach lets you see the behavior of the sequence for the first 10 steps, laying the groundwork for understanding the overall pattern.

When doing these calculations, keep in mind the effect of the alternating sign caused by \((-1)^{n}\), as it influences whether you'll add or subtract the \((n - 5)\) term. This factor can sometimes cause confusion, so pay special attention to it to ensure you calculate the terms correctly.

Identifying Duplicate Terms

Upon calculating the first 10 terms of the sequence, it's important to compare them to identify any duplicates. Duplicate terms are values that appear more than once in the list of computed terms. This step is important to understand the repetition and potential patterns within the sequence.

To identify duplicates, simply list out the calculated terms and scan through to see if any number is listed more than once. For instance, if both \(x_{3}\) and \(x_{7}\) equal 12, then 12 is a duplicate term in the sequence. Recognizing duplicates can help simplify problems and lead to insights into the structure of the sequence.

Thoroughly checking each term against all the others can be time-consuming, so some students find it helpful to arrange the terms in numerical order or use a table format to aid in quick identification of any repeating numbers.

Frequency of Sequence Terms

Determining the frequency of sequence terms involves counting how many times a particular term appears within the sequence. After identifying any duplicates, you'll want to ascertain if any of the terms appear not just twice, but even three times or more.

To check for the term frequency, create a tally chart or maintain a count of how many times each term occurs as you work through the computed values. This process requires careful attention to detail to ensure accuracy.

The frequency analysis adds another layer of understanding to the sequence's behavior. In our exercise, if a term appears three times within the first 10 terms, it indicates a strong recurring pattern that could be relevant in more advanced analysis or extrapolating the sequence further. If no such terms exist, it can imply more complexity or a lack of repetition in the sequence structure.